Trigonometry

1. Trigonometric ratios

Finding a trigonometric ratio

Answer the following questions about this right-angled triangle:

  1. Complete the definition of cosθ using the dropdown menus below:

    Answer:

    cosθ is defined as the over the .

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


    STEP: Complete the definition of cosθ
    [−2 points ⇒ 0 / 2 points left]

    We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

    sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

    For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

    Therefore, cosθ is defined as the adjacent over the hypotenuse.


    Submit your answer as: and
  2. What is the value of the ratio cosθ for the given right-angled triangle?

    Answer: cosθ= .
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Identify which two sides of the triangle correspond to the sides you used in the definition of cosθ in Question 1, and take the ratio of these two sides.


    STEP: Take the ratio of the correct two sides of the triangle
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to determine cosθ for the given right-angled triangle.

    From Question 1, we know that cosθ is defined as the adjacent over the hypotenuse. We need to identify which sides of the triangle are the adjacent and hypotenuse, and take the ratio of their lengths.

    For the given triangle, the adjacent has length 3 and the hypotenuse has length 5.

    cosθ=adjacenthypotenuse=35

    Therefore, cosθ is 35.


    Submit your answer as:

A connection between sine and cosine

The diagram below shows a right-angled triangle, with acute angles labelled x and y, xy. The sides of the triangle are labelled A, B, and C.

Because this is a right-angled triangle, the following statement is true:

sinx=cosy

Why is this statement true?

Answer:

sinx=cosy because .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the sides of the triangle needed for the ratios sinx and cosy. What do these sides have in common?


STEP: Identify the sides used for sinx and cosy
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios sine (sin) and cosine (cos) tell us about the ratio of different sides relative to a particular angle θ in a right-angled triangle.

sinθ=oppositehypotenusecosθ=adjacenthypotenuse

Deciding which side of the triangle is labelled as the 'opposite' side or the 'adjacent' side depends on the location of the angle that we are working with. The opposite side is always directly across the triangle from the angle. The adjacent side is the side next to the angle. (The hypotenuse is always the side across from the right-angle.)

For sinx, we need to identify the side opposite the angle x. For cosy, we need to find the side adjacent to the angle y. This is shown with the red and blue arrows below.

Looking at the picture, we can see that the arrows are pointing to the same side! Since this side is the numerator of both of the ratios, and the hypotenuse is the denominator for both of them, we now know

sinx=BC=cosy
This rule is always true for complementary angles (angles that add up to 90°) in a right-angled triangle. This is also why the cosine ratio is called cosine: it is the COmplementary function of SINE!

Therefore, the correct answer is sinx=cosy because the side opposite x is the side adjacent to y.


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Opposite and adjacent sides: connecting to angles

The diagram below shows a right-angled triangle with sides Q, R, and S, and two acute angles: ω, and θ.

Which side is adjacent to the angle θ?

Answer: The side adjacent to angle θ is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

The side that is adjacent to the angle θ is situated next to the angle θ.


STEP: Identify the side adjacent to the angle θ
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The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is the side S.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle θ, so the angle of interest is angle θ. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle θ is the adjacent side.
  • The remaining side that is faraway from the angle θ is the opposite side.

There is another acute angle in the triangle, which we can call ω. If we change the angle of interest to ω, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: The sides R and Q can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle θ is R.


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Working with trigonometric ratios

Determine the value of tanθ in the following triangle.

Select an answer from the following list:

A 32
B 22
C 12
D 1
E 33
F 3
Answer: tanθ=
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the opposite, adjacent, and hypotenuse sides of the triangle. Then use the definition of tanθ to get the answer.


STEP: Use the definition of tanθ to get the answer
[−1 point ⇒ 0 / 1 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of tanθ. The definition of the tangent ratio is:

tanθ=oppositeadjacent

We need to find the lengths for the opposite side and the adjacent side. The opposite and the adjacents sides both depend on the location of angle θ. The purple arrows in the diagram below show those two sides. The opposite side is across the triangle from θ, while the adjacent side is next to θ.

In this case the length of the opposite side is 43 and the length of the adjacent side is 4. Therefore,

tanθ=oppositeadjacent=434=3
NOTE: The hypotenuse of the triangle is greyed out in the diagram above. That is because it is not needed for this question: the tangent ratio does not care about that side of the triangle.

The correct choice from the list is F.


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Right triangles: opposite and adjacent sides

The diagram below shows a right-angled triangle with sides labelled Side 1, Side 2, and Side 3. There is also an acute angle θ.

Which sides are adjacent and opposite to the angle θ?

Answer:

The side adjacent to angle θ is .

The side opposite to angle θ is .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The side that is adjacent to the angle θ is next to the angle θ. The side that is opposite to the angle θ is across from the angle θ.


STEP: Identify the sides opposite and adjacent to the angle θ
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The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is Side 2.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle θ, so the angle of interest is angle θ. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle θ is the adjacent side.
  • The remaining side that is far away from the angle θ is the opposite side.

There is another acute angle in the triangle, which we can call ϕ. If we change the angle of interest to ϕ, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: Side 3 and Side 1 can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle θ is Side 3.

The side opposite to angle θ is Side 1.


Submit your answer as: and

Calculation of the area of a triangle

A triangle ABC is shown in the figure below. The length of AC¯ is 4 units. The size of the angle BA^C is 50°.

  1. Calculate the lengths of the line segments AB¯ and BC¯. You will need to use your knowledge of trigonometric ratios for these calculations. Round your answers to 2 decimal places.

    Answer:

    AB¯= units
    BC¯= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Use the definitions of the sine and cosine functions.


    STEP: Choose the correct trigonometric ratios
    [−2 points ⇒ 2 / 4 points left]

    We are given the following information:

    • The triangle has a right angle (AB^C) (shown in the diagram)
    • AC¯=4 units
    • BA^C=50°

    This means we can use the cosine (cos) and sine (sin) functions to detemine the lengths of the line segments AB¯ and BC¯. The cos and sin of an angle, for example, θ are defined as follows:

    cosθ=adjacenthypotenusesinθ=oppositehypotenuse

    Line segment AB¯ is adjacent to the angle BA^C and line segment BC¯ is opposite to the angle BA^C. Line segment AC¯ is the hypotenuse (it is opposite the right angle AB^C). We know the length of AC¯ and the size of angle BA^C, so we can calculate the unkown line segments.

    Therefore, we will use the cosine function to calculate the length of AB¯ and the sine function to calculate the length of BC¯.


    STEP: Use the trigonometric ratios to find the sides of the triangle
    [−2 points ⇒ 0 / 4 points left]

    We will now calculate the lengths of AB¯ and BC¯.

    For AB¯:

    cosθ=adjacenthypotenusecos(BA^C)==AB¯=AC¯AB¯=cos(BA^C)×AC¯=cos50°×4=2,57115...2,57 units

    For BC¯:

    sinθ=oppositehypotenusesin(BA^C)==BC¯=AC¯BC¯=sin(BA^C)×AC¯=sin50°×4=3,06417...3,06 units

    Therefore, the length of AB¯=2,57 units and the length of BC¯=3,06 units.


    Submit your answer as: and
  2. Which formula from the table below would you use to calculate the area of the triangle ABC?

    Choice Formula
    A AB¯×BC¯×CA¯
    B 12×AB¯×BC¯×CA¯
    C AB¯×BC¯
    D 12×AB¯×BC¯
    E π×(AB¯)2
    Answer: The correct formula is choice .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Consider a right triangle to be a result of dividing a rectangle into two parts along the diagonal. Do you remember the formula for the area of a rectangle?


    STEP: Determine the formula using the relationship between a right triangle and a rectangle
    [−1 point ⇒ 0 / 1 points left]

    We can consider the triangle ABC part of a rectangle ABCD, as shown below:

    The area of the rectangle ABCD is simply length multiplied by the breadth, i.e. AB¯×BC¯ or CD¯×AD¯. Notice, the area of the triangle ABC is exactly half of the area of the rectangle ABCD. Therefore the area of the triangle ABC is equal to 12×AB¯×BC¯.

    You can determine the area of any right angled triangle by multiplying the two sides adjacent to the right angle, and dividing by 2 (or multiplying by a half).

    Therefore, the correct formula is choice D.


    Submit your answer as:
  3. Using the unrounded answers from the first question, calculate the area of the triangle ABC. Round your answers to 2 decimal places.

    Answer: Area of triangle ABC= units2
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the formula for the area of a triangle.


    STEP: Use the formula for the area of a triangle
    [−2 points ⇒ 0 / 2 points left]

    From the previous question, we have found that the formula for the area of the triangle ABC is 12×AB¯×BC¯.

    From Question 1, we found that AB¯=2,57115... units and BC¯=3,06417... units.

    The area of the triangle ABC:

    Area of triangle ABC=12×AB¯×BC¯=12×2,57115...×3,06417...=3,93923...3,94 units2

    Therefore, the area of the triangle ABC is 3,94 units2.


    Submit your answer as:

Identify and use the correct trigonometric ratio

Study the given right-angled triangle ΔNSQ with N=90° and answer the following question. The angle given in the diagram is: NSQ=24,66°. The side given is: SQ¯=11.

Calculate the length of side SN¯. Round your answer to two decimal places if appropriate.

Answer: SN¯=
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is figure out which trigonometric ratio to use: sine, cosine, or tangent.


STEP: Find the correct trigonometric ratio to solve for the unkown
[−2 points ⇒ 2 / 4 points left]

From the question we know the side SQ¯=11 and the angle NSQ=24,66°. And we need to find SN¯. These sides are the hypotenuse and the side adjacent to angle NSQ. The trigonometric ratio which links the adjacent side to the hypotenuse is cosine. We can write an equation based on the cosine ratio as show here, and we can call the unknown side y.

cos24,66°=SN¯SQ¯cos24,66°=ySQ¯

STEP: Solve the trigonometric equation
[−2 points ⇒ 0 / 4 points left]

Now multiply both sides of the equation by the denominator.

cos24,66°=y11,011,0cos24,66°=yy=11,0cos24,66°y=9,99679...y10

Remember that the instructions say to round the answer to the second decimal place, as shown in the last step above.

The length of SN¯ is 10.


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Trigonometry: using a calculator

Use your calculator to evaluate the following expression: cos5°.

INSTRUCTION: Round your answer to two decimal places.
Answer: cos5°
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to use your calculator to evaluate cos5°. If you are not sure about how to do that, refer to the manual of your calculator.


STEP: Use your calculator to evaluate cos5° and then round off to two decimal places
[−2 points ⇒ 0 / 2 points left]

To determine the answer, we must use a calculator.

cos5°=0,99619...1

The correct answer is 1.


Submit your answer as:

Trigonometric ratios

Which trigonometric ratio is defined as the ratio of the opposite and the hypotenuse?

Answer: The trigonometric ratio is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Identify the correct trigonometric ratio
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The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sine ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sine ratio.

NOTE: These ratios are only valid when the triangle has a right-angle. Otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios, otherwise you will not get the right answer.

Therefore, the ratio defined as the ratio of the opposite and the hypotenuse is sine.


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Trigonometric ratios: definitions

Complete the definition of sin using the dropdown menus below:

Answer:

Sin is defined as the over the .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Complete the definition of sin
[−2 points ⇒ 0 / 2 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sin ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sin ratio.

NOTE: These ratios are only valid when the triangle has a right-angle - otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios. Otherwise you will not get the right answer.

Therefore, sin is defined as the opposite over the hypotenuse.


Submit your answer as: and

Writing trigonometric ratios

The diagram below shows a right-angled triangle, with sides a, b, and c. The two acute angles are A and B. The diagram is not drawn to scale.

Determine the value of cosB for this triangle.

Answer: cosB=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Remember that the opposite and adjacent depend on the position of the angle B. Then use the definition of cosB to get the answer.


STEP: Evaluate cosB for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio cosB. The definition of cosine is:

cosB=adjacenthypotenuse

We need to find the lengths for the adjacent and the hypotenuse. The adjacent side depends on the location of the angle B. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the adjacent side. It is the side next to the angle B.

In this case, the length of the adjacent is c and the length of the hypotenuse is equal to b. Therefore:

cosB=adjacenthypotenuse=cb

Therefore, the answer is cosB=cb.


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Trigonometric ratios: opposite and adjacent sides

Consider the diagram below:

The following equation is based on the triangle above.

cos50°=R

What is the quantity represented by ?

Answer:

The missing quantity is .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Start by identifying the hypotenuse, adjacent, and opposite sides of the right-angled triangle given. Then use the following definition to answer the question:

cos50°=adjacenthypotenuse

STEP: Use the definition of cos 50° to get the answer
[−1 point ⇒ 0 / 1 points left]

We need to find the missing quantity in the equation:

cos50°=R

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is called the hypotenuse. In this question, the hypotenuse is the side R. The names of the other two sides depends on the angle of interest. In this question, the angle of interest is the one whose size is 50°. Using this angle:

  • side S is the adjacent side because it is next to (or adjacent to) the 50° angle.
  • side Q is the opposite side because it is across from (or opposite to) the 50° angle.

In order to define the cosine of an acute angle in a right-angled triangle, we make use of the adjacent and hypotenuse sides. In this case:

cos50°=adjacenthypotenuse

In the triangle diagram given, the label of the adjacent side is S and the label for the hypotenuse side is R. We have drawn circles around these labels in the diagram shown below:

NOTE: We have shaded the side Q because it is not required in the definition of the cosine of the angle 50°.

We can now use these labels to define cosine for the angle 50°:

cos50°=SR

Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity S.

The missing quantity is S.


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Trigonometric ratios

The following diagram shows a right-angled triangle. The sides are labelled with lengths 14, 16, and 0. The two acute angles are labelled α and β. The diagram is not drawn to scale.

Determine the value of tanβ in the following triangle.

INSTRUCTION: Give your answer in surd form, if necessary, and simplify your answer completely. Type sqrt( ) if you need to indicate a square root.
Answer: tanβ=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Then use the definition of tanβ to get the answer.


STEP: Determine the value of tanβ for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of tanβ.

To find tanβ, we need to identify the opposite and the adjacent. The opposite side is across the triangle from the angle β. The adjacent side is next to β. The arrows and circles below show the those two sides. The opposite side is across the triangle from β, and the adjacent side is next to β.

In this case, the length of the opposite is 14 and the length of the adjacent is equal to 0. Therefore:

tanβ=140=70=~

Therefore, tanβ=~.


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Trigonometric ratio definitions

The triangle MNP below is right-angled at M.

Use the triangle MNP to answer the following questions:

  1. What is the quantity represented by in the following equation?

    cosN^=NP
    Answer:

    The missing quantity is .

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the angle N^ to identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. After that, use the following definition to answer the question:

    cosN^=adjacenthypotenuse

    STEP: Use the definition of cosN^ to get the answer
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    cosN^=NP

    The names of the sides of a right-angled triangle are hypotenuse, adjacent, and opposite sides. The side that does not touch the angle M^ is the hypotenuse. In this question, the hypotenuse is the side NP. The names of the other two sides depend on the angle of interest. In this question, the angle of interest is angle N^. Using this angle:

    • side MN is the adjacent side because it is next to (or adjacent to) the angle N^.
    • side MP is the opposite side because it is across from (or opposite to) the angle N^.

    To define cosine we make use of the sides: adjacent and hypotenuse. The definition of cosine for the angle N^ is:

    cosN^=adjacenthypotenuse

    From the triangle diagram, the label for the adjacent side is MN, and the label for the hypotenuse side is NP.

    Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity MN. We can now complete the equation:

    cosN^=MNNP
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the cosine of the angle N^. These are: MN and NP.

    The opposite side, MP, has been shaded out because it is not needed for the cosine ratio.

    The missing quantity is MN.


    Submit your answer as:
  2. From the drop-down list below, choose the quantity that would make the following equation complete.

    P^=MNNP
    Answer: The missing quantity is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    The missing quantity is a trigonometric function. Using the angle P^, identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. Then use any of the following definitions to answer the question:

    cosP^=adjacenthypotenusesinP^=oppositehypotenusetanP^=oppositeadjacent

    STEP: Identify the trigonometric ratio of the angle P^ equal to the ratio MNNP
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    P^=MNNP

    The missing quantity is a trigonometric function. Trigonometric functions of acute angles of a right-angled triangle are equal to the ratio of two of the sides. The definitions of three trigonometric functions of the angle P^ using the names of the sides, are:

    cosP^=adjacenthypotenusesinP^=oppositehypotenusetanP^=oppositeadjacent

    We now need to identify the names of the sides: MN and NP. Then we will identify the missing trigonometric function.

    For the angle P^, the adjacent is the side MP, the opposite is side MN, and the hypotenuse is side NP. The equation in the question statement has the ratio of the opposite side to the hypotenuse side. Using the equations of the definitions of the three trigonometric ratios above, we find that the missing quantity is sin. The completed equation is:

    sinP^=MNNP
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the sine of the angle P^. These are: the opposite side MN and the hypotenuse side NP.

    The missing quantity is sin.


    Submit your answer as:

Evaluating trigonometric ratios

The diagram below shows a right-angled triangle with sides of length 18, 24, and 30. The two acute angles are labelled x and y.

Determine the value of siny for this triangle.

INSTRUCTION: Give your answer as a simplified fraction.
Answer: siny=
fraction
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Start by writing down the definition of the sin ratio. Then identify which sides of the triangle correspond to the sides in the definition to find the answer.


STEP: Determine the value of siny for the given triangle
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We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio siny.

The definition of siny is

siny=oppositehypotenuse

So we need to identify the opposite and hypotenuse sides for the angle y. The opposite side depends on the location of the angle y. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the opposite side. It is the side across the triangle from the angle y.

In this case, the length of the opposite is 24 and the length of the hypotenuse is 30. Therefore:

siny=oppositehypotenuse=2430=45

Therefore, the answer is siny=45.


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Exercises

Finding a trigonometric ratio

Answer the following questions about this right-angled triangle:

  1. Complete the definition of tanθ using the dropdown menus below:

    Answer:

    tanθ is defined as the ratio of the and the .

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


    STEP: Complete the definition of tanθ
    [−2 points ⇒ 0 / 2 points left]

    We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

    sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

    For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

    Therefore, tanθ is defined as the ratio of the opposite and the adjacent.


    Submit your answer as: and
  2. What is the value of the ratio tanθ for the given right-angled triangle?

    Answer: tanθ= .
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Identify which two sides of the triangle correspond to the sides you used in the definition of tanθ in Question 1, and take the ratio of these two sides.


    STEP: Take the ratio of the correct two sides of the triangle
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to determine tanθ for the given right-angled triangle.

    From Question 1, we know that tanθ is defined as the ratio of the opposite and the adjacent. We need to identify which sides of the triangle are the opposite and adjacent, and take the ratio of their lengths.

    For the given triangle, the opposite has length 12 and the adjacent has length 5.

    tanθ=oppositeadjacent=125

    Therefore, tanθ is 125.


    Submit your answer as:

Finding a trigonometric ratio

Answer the following questions about this right-angled triangle:

  1. Complete the definition of tanθ using the dropdown menus below:

    Answer:

    tanθ is defined as the over the .

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


    STEP: Complete the definition of tanθ
    [−2 points ⇒ 0 / 2 points left]

    We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

    sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

    For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

    Therefore, tanθ is defined as the opposite over the adjacent.


    Submit your answer as: and
  2. What is the value of the ratio tanθ for the given right-angled triangle?

    Answer: tanθ= .
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Identify which two sides of the triangle correspond to the sides you used in the definition of tanθ in Question 1, and take the ratio of these two sides.


    STEP: Take the ratio of the correct two sides of the triangle
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to determine tanθ for the given right-angled triangle.

    From Question 1, we know that tanθ is defined as the opposite over the adjacent. We need to identify which sides of the triangle are the opposite and adjacent, and take the ratio of their lengths.

    For the given triangle, the opposite has length 4 and the adjacent has length 3.

    tanθ=oppositeadjacent=43

    Therefore, tanθ is 43.


    Submit your answer as:

Finding a trigonometric ratio

Answer the following questions about this right-angled triangle:

  1. Complete the definition of tanθ using the dropdown menus below:

    Answer:

    tanθ is defined as the over the .

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


    STEP: Complete the definition of tanθ
    [−2 points ⇒ 0 / 2 points left]

    We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

    sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

    For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

    Therefore, tanθ is defined as the opposite over the adjacent.


    Submit your answer as: and
  2. What is the value of the ratio tanθ for the given right-angled triangle?

    Answer: tanθ= .
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Identify which two sides of the triangle correspond to the sides you used in the definition of tanθ in Question 1, and take the ratio of these two sides.


    STEP: Take the ratio of the correct two sides of the triangle
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to determine tanθ for the given right-angled triangle.

    From Question 1, we know that tanθ is defined as the opposite over the adjacent. We need to identify which sides of the triangle are the opposite and adjacent, and take the ratio of their lengths.

    For the given triangle, the opposite has length 3 and the adjacent has length 4.

    tanθ=oppositeadjacent=34

    Therefore, tanθ is 34.


    Submit your answer as:

A connection between sine and cosine

The diagram below shows a right-angled triangle, with acute angles labelled a and b, ab. The sides of the triangle are labelled A, B, and C.

Because this is a right-angled triangle, the following statement is true:

sina=cosb

Why is this statement true?

Answer:

sina=cosb because .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the sides of the triangle needed for the ratios sina and cosb. What do these sides have in common?


STEP: Identify the sides used for sina and cosb
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios sine (sin) and cosine (cos) tell us about the ratio of different sides relative to a particular angle θ in a right-angled triangle.

sinθ=oppositehypotenusecosθ=adjacenthypotenuse

Deciding which side of the triangle is labelled as the 'opposite' side or the 'adjacent' side depends on the location of the angle that we are working with. The opposite side is always directly across the triangle from the angle. The adjacent side is the side next to the angle. (The hypotenuse is always the side across from the right-angle.)

For sina, we need to identify the side opposite the angle a. For cosb, we need to find the side adjacent to the angle b. This is shown with the red and blue arrows below.

Looking at the picture, we can see that the arrows are pointing to the same side! Since this side is the numerator of both of the ratios, and the hypotenuse is the denominator for both of them, we now know

sina=BC=cosb
This rule is always true for complementary angles (angles that add up to 90°) in a right-angled triangle. This is also why the cosine ratio is called cosine: it is the COmplementary function of SINE!

Therefore, the correct answer is sina=cosb because the side opposite a is the side adjacent to b.


Submit your answer as:

A connection between sine and cosine

The diagram below shows a right-angled triangle, with acute angles labelled ϕ and λ, ϕλ. The sides of the triangle are labelled R, S, and T.

Because this is a right-angled triangle, the following statement is true:

sinλ=cosϕ

Why is this statement true?

Answer:

sinλ=cosϕ because .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the sides of the triangle needed for the ratios sinλ and cosϕ. What do these sides have in common?


STEP: Identify the sides used for sinλ and cosϕ
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios sine (sin) and cosine (cos) tell us about the ratio of different sides relative to a particular angle θ in a right-angled triangle.

sinθ=oppositehypotenusecosθ=adjacenthypotenuse

Deciding which side of the triangle is labelled as the 'opposite' side or the 'adjacent' side depends on the location of the angle that we are working with. The opposite side is always directly across the triangle from the angle. The adjacent side is the side next to the angle. (The hypotenuse is always the side across from the right-angle.)

For sinλ, we need to identify the side opposite the angle λ. For cosϕ, we need to find the side adjacent to the angle ϕ. This is shown with the red and blue arrows below.

Looking at the picture, we can see that the arrows are pointing to the same side! Since this side is the numerator of both of the ratios, and the hypotenuse is the denominator for both of them, we now know

sinλ=TS=cosϕ
This rule is always true for complementary angles (angles that add up to 90°) in a right-angled triangle. This is also why the cosine ratio is called cosine: it is the COmplementary function of SINE!

Therefore, the correct answer is sinλ=cosϕ because the side opposite λ is the side adjacent to ϕ.


Submit your answer as:

A connection between sine and cosine

The diagram below shows a right-angled triangle, with acute angles labelled a and b, ab. The sides of the triangle are labelled R, S, and T.

Because this is a right-angled triangle, the following statement is true:

sinb=cosa

Why is this statement true?

Answer:

sinb=cosa because .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the sides of the triangle needed for the ratios sinb and cosa. What do these sides have in common?


STEP: Identify the sides used for sinb and cosa
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios sine (sin) and cosine (cos) tell us about the ratio of different sides relative to a particular angle θ in a right-angled triangle.

sinθ=oppositehypotenusecosθ=adjacenthypotenuse

Deciding which side of the triangle is labelled as the 'opposite' side or the 'adjacent' side depends on the location of the angle that we are working with. The opposite side is always directly across the triangle from the angle. The adjacent side is the side next to the angle. (The hypotenuse is always the side across from the right-angle.)

For sinb, we need to identify the side opposite the angle b. For cosa, we need to find the side adjacent to the angle a. This is shown with the red and blue arrows below.

Looking at the picture, we can see that the arrows are pointing to the same side! Since this side is the numerator of both of the ratios, and the hypotenuse is the denominator for both of them, we now know

sinb=TS=cosa
This rule is always true for complementary angles (angles that add up to 90°) in a right-angled triangle. This is also why the cosine ratio is called cosine: it is the COmplementary function of SINE!

Therefore, the correct answer is sinb=cosa because the side opposite b is the side adjacent to a.


Submit your answer as:

Opposite and adjacent sides: connecting to angles

The diagram below shows a right-angled triangle with sides X, Y, and Z, and two acute angles: θ, and δ.

Which side is adjacent to the angle θ?

Answer: The side adjacent to angle θ is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

The side that is adjacent to the angle θ is situated next to the angle θ.


STEP: Identify the side adjacent to the angle θ
[−1 point ⇒ 0 / 1 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is the side Z.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle θ, so the angle of interest is angle θ. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle θ is the adjacent side.
  • The remaining side that is faraway from the angle θ is the opposite side.

There is another acute angle in the triangle, which we can call δ. If we change the angle of interest to δ, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: The sides X and Y can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle θ is X.


Submit your answer as:

Opposite and adjacent sides: connecting to angles

The diagram below shows a right-angled triangle with sides X, Y, and Z, and two acute angles: α, and ω.

Which side is opposite to the angle α?

Answer: The side opposite to angle α is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

The side that is opposite to the angle α is situated across from the angle α.


STEP: Identify the side opposite to the angle α
[−1 point ⇒ 0 / 1 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is the side Y.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle α, so the angle of interest is angle α. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle α is the adjacent side.
  • The remaining side that is faraway from the angle α is the opposite side.

There is another acute angle in the triangle, which we can call ω. If we change the angle of interest to ω, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: The sides X and Z can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side opposite to angle α is Z.


Submit your answer as:

Opposite and adjacent sides: connecting to angles

The diagram below shows a right-angled triangle with sides Q, R, and S, and two acute angles: β, and α.

Which side is opposite to the angle α?

Answer: The side opposite to angle α is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

The side that is opposite to the angle α is situated across from the angle α.


STEP: Identify the side opposite to the angle α
[−1 point ⇒ 0 / 1 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is the side S.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle α, so the angle of interest is angle α. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle α is the adjacent side.
  • The remaining side that is faraway from the angle α is the opposite side.

There is another acute angle in the triangle, which we can call β. If we change the angle of interest to β, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: The sides R and Q can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side opposite to angle α is Q.


Submit your answer as:

Working with trigonometric ratios

Determine the value of tanθ in the following triangle.

Select an answer from the following list:

A 32
B 22
C 12
D 1
E 33
F 3
Answer: tanθ=
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the opposite, adjacent, and hypotenuse sides of the triangle. Then use the definition of tanθ to get the answer.


STEP: Use the definition of tanθ to get the answer
[−1 point ⇒ 0 / 1 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of tanθ. The definition of the tangent ratio is:

tanθ=oppositeadjacent

We need to find the lengths for the opposite side and the adjacent side. The opposite and the adjacents sides both depend on the location of angle θ. The purple arrows in the diagram below show those two sides. The opposite side is across the triangle from θ, while the adjacent side is next to θ.

In this case the length of the opposite side is 52 and the length of the adjacent side is 52. Therefore,

tanθ=oppositeadjacent=5252=1
NOTE: The hypotenuse of the triangle is greyed out in the diagram above. That is because it is not needed for this question: the tangent ratio does not care about that side of the triangle.

The correct choice from the list is D.


Submit your answer as:

Working with trigonometric ratios

Determine the value of tanθ in the following triangle.

Select an answer from the following list:

A 32
B 22
C 12
D 1
E 33
F 3
Answer: tanθ=
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the opposite, adjacent, and hypotenuse sides of the triangle. Then use the definition of tanθ to get the answer.


STEP: Use the definition of tanθ to get the answer
[−1 point ⇒ 0 / 1 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of tanθ. The definition of the tangent ratio is:

tanθ=oppositeadjacent

We need to find the lengths for the opposite side and the adjacent side. The opposite and the adjacents sides both depend on the location of angle θ. The purple arrows in the diagram below show those two sides. The opposite side is across the triangle from θ, while the adjacent side is next to θ.

In this case the length of the opposite side is 63 and the length of the adjacent side is 18. Therefore,

tanθ=oppositeadjacent=6318=33
NOTE: The hypotenuse of the triangle is greyed out in the diagram above. That is because it is not needed for this question: the tangent ratio does not care about that side of the triangle.

The correct choice from the list is E.


Submit your answer as:

Working with trigonometric ratios

Determine the value of sinθ in the following triangle.

Select an answer from the following list:

A 32
B 22
C 12
D 1
E 33
F 3
Answer: sinθ=
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Identify the opposite, adjacent, and hypotenuse sides of the triangle. Then use the definition of sinθ to get the answer.


STEP: Use the definition of sinθ to get the answer
[−1 point ⇒ 0 / 1 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of sinθ. The definition of the sine ratio is:

sinθ=oppositehypotenuse

We need to find the lengths for the opposite side and the hypotenuse. The opposite side depends on the location of angle θ. (The hypotenuse is always the side of the triangle across from the right angle.) The arrow and circle below show the opposite side. It is the side across from the angle θ.

In this case the length of the opposite side is 62 and the length of the hypotenuse is 12. Therefore,

sinθ=oppositehypotenuse=6212=22
NOTE: The adjacent side of the triangle is greyed out in the diagram above. That is because it is not needed for this question: the sine ratio does not care about that side of the triangle.

The correct choice from the list is B.


Submit your answer as:

Right triangles: opposite and adjacent sides

The diagram below shows a right-angled triangle with sides labelled Side 1, Side 2, and Side 3. There is also an acute angle ϕ.

Which sides are adjacent and opposite to the angle ϕ?

Answer:

The side adjacent to angle ϕ is .

The side opposite to angle ϕ is .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The side that is adjacent to the angle ϕ is next to the angle ϕ. The side that is opposite to the angle ϕ is across from the angle ϕ.


STEP: Identify the sides opposite and adjacent to the angle ϕ
[−2 points ⇒ 0 / 2 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is Side 3.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle ϕ, so the angle of interest is angle ϕ. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle ϕ is the adjacent side.
  • The remaining side that is far away from the angle ϕ is the opposite side.

There is another acute angle in the triangle, which we can call α. If we change the angle of interest to α, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: Side 2 and Side 1 can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle ϕ is Side 2.

The side opposite to angle ϕ is Side 1.


Submit your answer as: and

Right triangles: opposite and adjacent sides

The diagram below shows a right-angled triangle with sides labelled Side 1, Side 2, and Side 3. There is also an acute angle ω.

Which sides are adjacent and opposite to the angle ω?

Answer:

The side adjacent to angle ω is .

The side opposite to angle ω is .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The side that is adjacent to the angle ω is next to the angle ω. The side that is opposite to the angle ω is across from the angle ω.


STEP: Identify the sides opposite and adjacent to the angle ω
[−2 points ⇒ 0 / 2 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is Side 3.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle ω, so the angle of interest is angle ω. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle ω is the adjacent side.
  • The remaining side that is far away from the angle ω is the opposite side.

There is another acute angle in the triangle, which we can call δ. If we change the angle of interest to δ, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: Side 2 and Side 1 can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle ω is Side 2.

The side opposite to angle ω is Side 1.


Submit your answer as: and

Right triangles: opposite and adjacent sides

The diagram below shows a right-angled triangle with sides labelled Side 1, Side 2, and Side 3. There is also an acute angle θ.

Which sides are adjacent and opposite to the angle θ?

Answer:

The side adjacent to angle θ is .

The side opposite to angle θ is .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The side that is adjacent to the angle θ is next to the angle θ. The side that is opposite to the angle θ is across from the angle θ.


STEP: Identify the sides opposite and adjacent to the angle θ
[−2 points ⇒ 0 / 2 points left]

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is the hypotenuse. The hypotenuse is also the longest side of a right-angled triangle. In this question, the hypotenuse is Side 2.

We label the other two sides according to the location of the angle of interest. This question asked about the sides adjacent and opposite to the angle θ, so the angle of interest is angle θ. Using this angle, we label the other two sides in this manner:

  • The side that touches the angle θ is the adjacent side.
  • The remaining side that is far away from the angle θ is the opposite side.

There is another acute angle in the triangle, which we can call δ. If we change the angle of interest to δ, the labels opposite and adjacent will change. However, the hypotenuse remains the same.

We can summarise the above information using the diagram below:

NOTE: Side 3 and Side 1 can each be the opposite side or the adjacent side. It depends on the angle of interest.

The side adjacent to angle θ is Side 3.

The side opposite to angle θ is Side 1.


Submit your answer as: and

Calculation of the area of a triangle

A triangle ABC is shown in the figure below. The length of AC¯ is 4,4 units. The size of the angle BA^C is 34°.

  1. Work out the lengths of the line segments AB¯ and BC¯. You will need to use your knowledge of trigonometric ratios for these calculations. Round your answers to 2 decimal places.

    Answer:

    AB¯= units
    BC¯= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Use the definitions of the sine and cosine functions.


    STEP: Choose the correct trigonometric ratios
    [−2 points ⇒ 2 / 4 points left]

    We are given the following information:

    • The triangle has a right angle (AB^C) (shown in the diagram)
    • AC¯=4,4 units
    • BA^C=34°

    This means we can use the cosine (cos) and sine (sin) functions to detemine the lengths of the line segments AB¯ and BC¯. The cos and sin of an angle, for example, θ are defined as follows:

    cosθ=adjacenthypotenusesinθ=oppositehypotenuse

    Line segment AB¯ is adjacent to the angle BA^C and line segment BC¯ is opposite to the angle BA^C. Line segment AC¯ is the hypotenuse (it is opposite the right angle AB^C). We know the length of AC¯ and the size of angle BA^C, so we can calculate the unkown line segments.

    Therefore, we will use the cosine function to calculate the length of AB¯ and the sine function to calculate the length of BC¯.


    STEP: Use the trigonometric ratios to find the sides of the triangle
    [−2 points ⇒ 0 / 4 points left]

    We will now calculate the lengths of AB¯ and BC¯.

    For AB¯:

    cosθ=adjacenthypotenusecos(BA^C)==AB¯=AC¯AB¯=cos(BA^C)×AC¯=cos34°×4,4=3,64776...3,65 units

    For BC¯:

    sinθ=oppositehypotenusesin(BA^C)==BC¯=AC¯BC¯=sin(BA^C)×AC¯=sin34°×4,4=2,46044...2,46 units

    Therefore, the length of AB¯=3,65 units and the length of BC¯=2,46 units.


    Submit your answer as: and
  2. Which formula from the table below would you use to calculate the area of the triangle ABC?

    Choice Formula
    A 12×AB¯×BC¯×CA¯
    B AB¯×BC¯×CA¯
    C AB¯×BC¯
    D π×(AB¯)2
    E 12×AB¯×BC¯
    Answer: The correct formula is choice .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Consider a right triangle to be a result of dividing a rectangle into two parts along the diagonal. Do you remember the formula for the area of a rectangle?


    STEP: Determine the formula using the relationship between a right triangle and a rectangle
    [−1 point ⇒ 0 / 1 points left]

    We can consider the triangle ABC part of a rectangle ABCD, as shown below:

    The area of the rectangle ABCD is simply length multiplied by the breadth, i.e. AB¯×BC¯ or CD¯×AD¯. Notice, the area of the triangle ABC is exactly half of the area of the rectangle ABCD. Therefore the area of the triangle ABC is equal to 12×AB¯×BC¯.

    You can determine the area of any right angled triangle by multiplying the two sides adjacent to the right angle, and dividing by 2 (or multiplying by a half).

    Therefore, the correct formula is choice E.


    Submit your answer as:
  3. Using the unrounded answers from the first question, calculate the area of the triangle ABC. Round your answers to 2 decimal places.

    Answer: Area of triangle ABC= units2
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the formula for the area of a triangle.


    STEP: Use the formula for the area of a triangle
    [−2 points ⇒ 0 / 2 points left]

    From the previous question, we have found that the formula for the area of the triangle ABC is 12×AB¯×BC¯.

    From Question 1, we found that AB¯=3,64776... units and BC¯=2,46044... units.

    The area of the triangle ABC:

    Area of triangle ABC=12×AB¯×BC¯=12×3,64776...×2,46044...=4,48756...4,49 units2

    Therefore, the area of the triangle ABC is 4,49 units2.


    Submit your answer as:

Calculation of the area of a triangle

A triangle ABC is shown in the figure below. The length of AC¯ is 4,2 units. The size of the angle BA^C is 39°.

  1. Calculate the lengths of the line segments AB¯ and BC¯. You will need to use your knowledge of trigonometric ratios for these calculations. Round your answers to 2 decimal places.

    Answer:

    AB¯= units
    BC¯= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Use the definitions of the sine and cosine functions.


    STEP: Choose the correct trigonometric ratios
    [−2 points ⇒ 2 / 4 points left]

    We are given the following information:

    • The triangle has a right angle (AB^C) (shown in the diagram)
    • AC¯=4,2 units
    • BA^C=39°

    This means we can use the cosine (cos) and sine (sin) functions to detemine the lengths of the line segments AB¯ and BC¯. The cos and sin of an angle, for example, θ are defined as follows:

    cosθ=adjacenthypotenusesinθ=oppositehypotenuse

    Line segment AB¯ is adjacent to the angle BA^C and line segment BC¯ is opposite to the angle BA^C. Line segment AC¯ is the hypotenuse (it is opposite the right angle AB^C). We know the length of AC¯ and the size of angle BA^C, so we can calculate the unkown line segments.

    Therefore, we will use the cosine function to calculate the length of AB¯ and the sine function to calculate the length of BC¯.


    STEP: Use the trigonometric ratios to find the sides of the triangle
    [−2 points ⇒ 0 / 4 points left]

    We will now calculate the lengths of AB¯ and BC¯.

    For AB¯:

    cosθ=adjacenthypotenusecos(BA^C)==AB¯=AC¯AB¯=cos(BA^C)×AC¯=cos39°×4,2=3,26401...3,26 units

    For BC¯:

    sinθ=oppositehypotenusesin(BA^C)==BC¯=AC¯BC¯=sin(BA^C)×AC¯=sin39°×4,2=2,64314...2,64 units

    Therefore, the length of AB¯=3,26 units and the length of BC¯=2,64 units.


    Submit your answer as: and
  2. Which formula from the table below would you use to calculate the area of the triangle ABC?

    Choice Formula
    A AB¯×BC¯
    B 12×AB¯×BC¯
    C AB¯×BC¯×CA¯
    D 12×AB¯×BC¯×CA¯
    E π×(AB¯)2
    Answer: The correct formula is choice .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Consider a right triangle to be a result of dividing a rectangle into two parts along the diagonal. Do you remember the formula for the area of a rectangle?


    STEP: Determine the formula using the relationship between a right triangle and a rectangle
    [−1 point ⇒ 0 / 1 points left]

    We can consider the triangle ABC part of a rectangle ABCD, as shown below:

    The area of the rectangle ABCD is simply length multiplied by the breadth, i.e. AB¯×BC¯ or CD¯×AD¯. Notice, the area of the triangle ABC is exactly half of the area of the rectangle ABCD. Therefore the area of the triangle ABC is equal to 12×AB¯×BC¯.

    You can determine the area of any right angled triangle by multiplying the two sides adjacent to the right angle, and dividing by 2 (or multiplying by a half).

    Therefore, the correct formula is choice B.


    Submit your answer as:
  3. Using the unrounded answers from the first question, work out the area of the triangle ABC. Round your answers to 2 decimal places.

    Answer: Area of triangle ABC= units2
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the formula for the area of a triangle.


    STEP: Use the formula for the area of a triangle
    [−2 points ⇒ 0 / 2 points left]

    From the previous question, we have found that the formula for the area of the triangle ABC is 12×AB¯×BC¯.

    From Question 1, we found that AB¯=3,26401... units and BC¯=2,64314... units.

    The area of the triangle ABC:

    Area of triangle ABC=12×AB¯×BC¯=12×3,26401...×2,64314...=4,31363...4,31 units2

    Therefore, the area of the triangle ABC is 4,31 units2.


    Submit your answer as:

Calculation of the area of a triangle

A triangle ABC is shown in the figure below. The length of AC¯ is 5 units. The size of the angle BA^C is 35°.

  1. Calculate the lengths of the line segments AB¯ and BC¯. You will need to use your knowledge of trigonometric ratios for these calculations. Round your answers to 2 decimal places.

    Answer:

    AB¯= units
    BC¯= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Use the definitions of the sine and cosine functions.


    STEP: Choose the correct trigonometric ratios
    [−2 points ⇒ 2 / 4 points left]

    We are given the following information:

    • The triangle has a right angle (AB^C) (shown in the diagram)
    • AC¯=5 units
    • BA^C=35°

    This means we can use the cosine (cos) and sine (sin) functions to detemine the lengths of the line segments AB¯ and BC¯. The cos and sin of an angle, for example, θ are defined as follows:

    cosθ=adjacenthypotenusesinθ=oppositehypotenuse

    Line segment AB¯ is adjacent to the angle BA^C and line segment BC¯ is opposite to the angle BA^C. Line segment AC¯ is the hypotenuse (it is opposite the right angle AB^C). We know the length of AC¯ and the size of angle BA^C, so we can calculate the unkown line segments.

    Therefore, we will use the cosine function to calculate the length of AB¯ and the sine function to calculate the length of BC¯.


    STEP: Use the trigonometric ratios to find the sides of the triangle
    [−2 points ⇒ 0 / 4 points left]

    We will now calculate the lengths of AB¯ and BC¯.

    For AB¯:

    cosθ=adjacenthypotenusecos(BA^C)==AB¯=AC¯AB¯=cos(BA^C)×AC¯=cos35°×5=4,09576...4,1 units

    For BC¯:

    sinθ=oppositehypotenusesin(BA^C)==BC¯=AC¯BC¯=sin(BA^C)×AC¯=sin35°×5=2,86788...2,87 units

    Therefore, the length of AB¯=4,1 units and the length of BC¯=2,87 units.


    Submit your answer as: and
  2. Which formula from the table below would you use to calculate the area of the triangle ABC?

    Choice Formula
    A AB¯×BC¯
    B 12×AB¯×BC¯
    C 12×AB¯×BC¯×CA¯
    D AB¯×BC¯×CA¯
    E π×(AB¯)2
    Answer: The correct formula is choice .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Consider a right triangle to be a result of dividing a rectangle into two parts along the diagonal. Do you remember the formula for the area of a rectangle?


    STEP: Determine the formula using the relationship between a right triangle and a rectangle
    [−1 point ⇒ 0 / 1 points left]

    We can consider the triangle ABC part of a rectangle ABCD, as shown below:

    The area of the rectangle ABCD is simply length multiplied by the breadth, i.e. AB¯×BC¯ or CD¯×AD¯. Notice, the area of the triangle ABC is exactly half of the area of the rectangle ABCD. Therefore the area of the triangle ABC is equal to 12×AB¯×BC¯.

    You can determine the area of any right angled triangle by multiplying the two sides adjacent to the right angle, and dividing by 2 (or multiplying by a half).

    Therefore, the correct formula is choice B.


    Submit your answer as:
  3. Using the unrounded answers from the first question, determine the area of the triangle ABC. Round your answers to 2 decimal places.

    Answer: Area of triangle ABC= units2
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the formula for the area of a triangle.


    STEP: Use the formula for the area of a triangle
    [−2 points ⇒ 0 / 2 points left]

    From the previous question, we have found that the formula for the area of the triangle ABC is 12×AB¯×BC¯.

    From Question 1, we found that AB¯=4,09576... units and BC¯=2,86788... units.

    The area of the triangle ABC:

    Area of triangle ABC=12×AB¯×BC¯=12×4,09576...×2,86788...=5,87307...5,87 units2

    Therefore, the area of the triangle ABC is 5,87 units2.


    Submit your answer as:

Identify and use the correct trigonometric ratio

The diagram below shows right-angled triangle ΔNPQ with N=90°. The sides given are: NP¯=6,93 and PQ¯=8.

Determine the value of NPQ^. Round your answer to two decimal places if appropriate.

Answer: NPQ= °
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

The first thing you need to do is figure out which trigonometric ratio to use: sine, cosine, or tangent.


STEP: Find the correct trigonometric ratio to solve for the unkown
[−2 points ⇒ 1 / 3 points left]

We need to determine the size of angle NPQ in the right-triangle, ΔNPQ. We need to determine which trigonometric ratio we can use to find NPQ. Side NP¯ is adjacent to the unknown angle. And side PQ¯ is the hypotenuse. The trigonometric ratio linking the hypotenuse to the adjacent side is the cosine ratio. So we can set up an equation with this information like below, using θ to represent the unknown angle.

cosθ=NP¯PQ¯

STEP: Solve the trigonometric equation
[−1 point ⇒ 0 / 3 points left]

Substitute the given values into the equation. Then solve the equation.

cosθ=6,938θ=cos1(6,938)NPQs=29,97425...°NPQ29,97°

Remember that the instructions say to round the answer to the second decimal place, as shown in the last step above.

The size of NPQ is 29,97°.


Submit your answer as:

Identify and use the correct trigonometric ratio

The diagram below shows right-angled triangle ΔABC with A=90°. The sides given are: AB¯=4,95 and BC¯=7.

Determine the value of ABC. Round your answer to two decimal places if appropriate.

Answer: ABC= °
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

The first thing you need to do is figure out which trigonometric ratio to use: sine, cosine, or tangent.


STEP: Find the correct trigonometric ratio to solve for the unkown
[−2 points ⇒ 1 / 3 points left]

We need to determine the size of angle ABC in the right-triangle, ΔABC. We need to determine which trigonometric ratio we can use to find ABC. Side AB¯ is adjacent to the unknown angle. And side BC¯ is the hypotenuse. The trigonometric ratio linking the hypotenuse to the adjacent side is the cosine ratio. So we can set up an equation with this information like below, using α to represent the unknown angle.

cosα=AB¯BC¯

STEP: Solve the trigonometric equation
[−1 point ⇒ 0 / 3 points left]

Substitute the given values into the equation. Then solve the equation.

cosα=4,957α=cos1(4,957)ABCs=44,99707...°ABC45,0°

Remember that the instructions say to round the answer to the second decimal place, as shown in the last step above.

The size of ABC is 45,0°.


Submit your answer as:

Identify and use the correct trigonometric ratio

Study the given right-angled triangle ΔXGV with X=90° and answer the following question. The angle given in the diagram is: GVX=51,34°. The side given is: GX¯=5.

Determine the length of side XV¯. Round your answer to two decimal places if appropriate.

Answer: XV¯=
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is figure out which trigonometric ratio to use: sine, cosine, or tangent.


STEP: Find the correct trigonometric ratio to solve for the unkown
[−2 points ⇒ 2 / 4 points left]

The question gives us the side GX¯=5 and GVX=51,34°. And we need to find side XV¯. Side GX¯ is opposite the given angle and side XV¯ is adjacent to the angle. This combination corresponds to the tangent ratio. Set up an equation with this information and call the missing side y.

tan51,34°=GX¯XV¯tan51,34°=GX¯y

STEP: Solve the trigonometric equation
[−2 points ⇒ 0 / 4 points left]

Substitute the given values into the equation we set up in the first step, and solve the equation.

tan51,34°=5yytan51,34°=5y=5tan51,34°y=4,00002...y4

Remember that the instructions say to round the answer to the second decimal place, as shown in the last step above.

The length of XV¯ is 4.


Submit your answer as:

Trigonometry: using a calculator

Use your calculator to evaluate the following expression: tan47°.

INSTRUCTION: Round your answer to two decimal places.
Answer: tan47°
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to use your calculator to evaluate tan47°. If you are not sure about how to do that, refer to the manual of your calculator.


STEP: Use your calculator to evaluate tan47° and then round off to two decimal places
[−2 points ⇒ 0 / 2 points left]

To determine the answer, we must use a calculator.

tan47°=1,07236...1,07

The correct answer is 1,07.


Submit your answer as:

Trigonometry: using a calculator

Use your calculator to evaluate the following expression: tan75°.

INSTRUCTION: Round your answer to two decimal places.
Answer: tan75°
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to use your calculator to evaluate tan75°. If you are not sure about how to do that, refer to the manual of your calculator.


STEP: Use your calculator to evaluate tan75° and then round off to two decimal places
[−2 points ⇒ 0 / 2 points left]

To determine the answer, we must use a calculator.

tan75°=3,73205...3,73

The correct answer is 3,73.


Submit your answer as:

Trigonometry: using a calculator

Use your calculator to evaluate the following expression: cos1°.

INSTRUCTION: Round your answer to two decimal places.
Answer: cos1°
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to use your calculator to evaluate cos1°. If you are not sure about how to do that, refer to the manual of your calculator.


STEP: Use your calculator to evaluate cos1° and then round off to two decimal places
[−2 points ⇒ 0 / 2 points left]

To determine the answer, we must use a calculator.

cos1°=0,99984...1

The correct answer is 1.


Submit your answer as:

Trigonometric ratios

Which trigonometric ratio is defined as the ratio of the opposite and the hypotenuse?

Answer: The trigonometric ratio is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Identify the correct trigonometric ratio
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sine ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sine ratio.

NOTE: These ratios are only valid when the triangle has a right-angle. Otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios, otherwise you will not get the right answer.

Therefore, the ratio defined as the ratio of the opposite and the hypotenuse is sine.


Submit your answer as:

Trigonometric ratios

Which trigonometric ratio is defined as the ratio of the opposite and the hypotenuse?

Answer: The trigonometric ratio is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Identify the correct trigonometric ratio
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sine ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sine ratio.

NOTE: These ratios are only valid when the triangle has a right-angle. Otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios, otherwise you will not get the right answer.

Therefore, the ratio defined as the ratio of the opposite and the hypotenuse is sine.


Submit your answer as:

Trigonometric ratios

Which trigonometric ratio is defined as the opposite over the hypotenuse?

Answer: The trigonometric ratio is .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Identify the correct trigonometric ratio
[−1 point ⇒ 0 / 1 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sine ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sine ratio.

NOTE: These ratios are only valid when the triangle has a right-angle. Otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios, otherwise you will not get the right answer.

Therefore, the ratio defined as the opposite over the hypotenuse is sine.


Submit your answer as:

Trigonometric ratios: definitions

Complete the definition of sin using the dropdown menus below:

Answer:

Sin is defined as the over the .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Complete the definition of sin
[−2 points ⇒ 0 / 2 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sin ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sin ratio.

NOTE: These ratios are only valid when the triangle has a right-angle - otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios. Otherwise you will not get the right answer.

Therefore, sin is defined as the opposite over the hypotenuse.


Submit your answer as: and

Trigonometric ratios: definitions

Complete the definition of tan using the dropdown menus below:

Answer:

Tan is defined as the ratio of the and the .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Complete the definition of tan
[−2 points ⇒ 0 / 2 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the tan ratio in purple: opposite and adjacent. The hypotenuse is shaded out because it is not needed for the tan ratio.

NOTE: These ratios are only valid when the triangle has a right-angle - otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios. Otherwise you will not get the right answer.

Therefore, tan is defined as the ratio of the opposite and the adjacent.


Submit your answer as: and

Trigonometric ratios: definitions

Complete the definition of sine using the dropdown menus below:

Answer:

Sine is defined as the ratio of the and the .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You can find the answer by reading about defining the trigonometric ratios in the Everything Maths Textbook.


STEP: Complete the definition of sine
[−2 points ⇒ 0 / 2 points left]

The trigonometric ratios tell us about two sides and an angle in a right-angled triangle.

We define the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) of an angle θ as follows:

sinθ=oppositehypotenusecosθ=adjacenthypotenusetanθ=oppositeadjacent

For a right-angled triangle, the side across the triangle from the right-angle is called the "hypotenuse". For either of the other angles in the triangle, the side across the triangle from the angle is labelled as the "opposite". And the side next to this angle is called the "adjacent" side.

The diagram shows the two sides we need for the sine ratio in purple: opposite and hypotenuse. The adjacent is shaded out because it is not needed for the sine ratio.

NOTE: These ratios are only valid when the triangle has a right-angle - otherwise the triangle would not have a hypotenuse. Always check that you have a right-angled triangle before you start using these trigonometric ratios. Otherwise you will not get the right answer.

Therefore, sine is defined as the ratio of the opposite and the hypotenuse.


Submit your answer as: and

Writing trigonometric ratios

The diagram below shows a right-angled triangle, with sides x, y, and z. The two acute angles are A and B. The diagram is not drawn to scale.

Determine the value of cosA for this triangle.

Answer: cosA=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Remember that the opposite and adjacent depend on the position of the angle A. Then use the definition of cosA to get the answer.


STEP: Evaluate cosA for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio cosA. The definition of cosine is:

cosA=adjacenthypotenuse

We need to find the lengths for the adjacent and the hypotenuse. The adjacent side depends on the location of the angle A. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the adjacent side. It is the side next to the angle A.

In this case, the length of the adjacent is x and the length of the hypotenuse is equal to z. Therefore:

cosA=adjacenthypotenuse=xz

Therefore, the answer is cosA=xz.


Submit your answer as:

Writing trigonometric ratios

The diagram below shows a right-angled triangle, with sides x, y, and z. The two acute angles are θ and λ. The diagram is not drawn to scale.

Determine the value of cosθ for this triangle.

Answer: cosθ=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Remember that the opposite and adjacent depend on the position of the angle θ. Then use the definition of cosθ to get the answer.


STEP: Evaluate cosθ for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio cosθ. The definition of cosine is:

cosθ=adjacenthypotenuse

We need to find the lengths for the adjacent and the hypotenuse. The adjacent side depends on the location of the angle θ. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the adjacent side. It is the side next to the angle θ.

In this case, the length of the adjacent is y and the length of the hypotenuse is equal to z. Therefore:

cosθ=adjacenthypotenuse=yz

Therefore, the answer is cosθ=yz.


Submit your answer as:

Writing trigonometric ratios

The diagram below shows a right-angled triangle, with sides p, q, and r. The two acute angles are A and B. The diagram is not drawn to scale.

Determine the value of sinA for this triangle.

Answer: sinA=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Remember that the opposite and adjacent depend on the position of the angle A. Then use the definition of sinA to get the answer.


STEP: Evaluate sinA for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio sinA. The definition of sine is:

sinA=oppositehypotenuse

We need to find the lengths for the opposite and the hypotenuse. The opposite side depends on the location of the angle A. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the opposite side. It is the side across the triangle from the angle A.

In this case, the length of the opposite is p and the length of the hypotenuse is equal to q. Therefore:

sinA=oppositehypotenuse=pq

Therefore, the answer is sinA=pq.


Submit your answer as:

Trigonometric ratios: opposite and adjacent sides

Consider the diagram below:

The following equation is based on the triangle above.

sin37°=S

What is the quantity represented by ?

Answer:

The missing quantity is .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Start by identifying the hypotenuse, adjacent, and opposite sides of the right-angled triangle given. Then use the following definition to answer the question:

sin37°=oppositehypotenuse

STEP: Use the definition of sin 37° to get the answer
[−1 point ⇒ 0 / 1 points left]

We need to find the missing quantity in the equation:

sin37°=S

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is called the hypotenuse. In this question, the hypotenuse is the side R. The names of the other two sides depends on the angle of interest. In this question, the angle of interest is the one whose size is 37°. Using this angle:

  • side Q is the adjacent side because it is next to (or adjacent to) the 37° angle.
  • side S is the opposite side because it is across from (or opposite to) the 37° angle.

In order to define the sine of an acute angle in a right-angled triangle, we make use of the opposite and hypotenuse sides. In this case:

sin37°=oppositehypotenuse

In the triangle diagram given, the label of the opposite side is S and the label for the hypotenuse side is R. We have drawn circles around these labels in the diagram shown below:

NOTE: We have shaded the side Q because it is not required in the definition of the sine of the angle 37°.

We can now use these labels to define sine for the angle 37°:

sin37°=SR

Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity R.

The missing quantity is R.


Submit your answer as:

Trigonometric ratios: opposite and adjacent sides

Consider the diagram below:

The following equation is based on the triangle above.

cos39°=S

What is the quantity represented by ?

Answer:

The missing quantity is .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Start by identifying the hypotenuse, adjacent, and opposite sides of the right-angled triangle given. Then use the following definition to answer the question:

cos39°=adjacenthypotenuse

STEP: Use the definition of cos 39° to get the answer
[−1 point ⇒ 0 / 1 points left]

We need to find the missing quantity in the equation:

cos39°=S

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is called the hypotenuse. In this question, the hypotenuse is the side R. The names of the other two sides depends on the angle of interest. In this question, the angle of interest is the one whose size is 39°. Using this angle:

  • side S is the adjacent side because it is next to (or adjacent to) the 39° angle.
  • side Q is the opposite side because it is across from (or opposite to) the 39° angle.

In order to define the cosine of an acute angle in a right-angled triangle, we make use of the adjacent and hypotenuse sides. In this case:

cos39°=adjacenthypotenuse

In the triangle diagram given, the label of the adjacent side is S and the label for the hypotenuse side is R. We have drawn circles around these labels in the diagram shown below:

NOTE: We have shaded the side Q because it is not required in the definition of the cosine of the angle 39°.

We can now use these labels to define cosine for the angle 39°:

cos39°=SR

Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity R.

The missing quantity is R.


Submit your answer as:

Trigonometric ratios: opposite and adjacent sides

Consider the diagram below:

The following equation is based on the triangle above.

sin69°=R

What is the quantity represented by ?

Answer:

The missing quantity is .

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Start by identifying the hypotenuse, adjacent, and opposite sides of the right-angled triangle given. Then use the following definition to answer the question:

sin69°=oppositehypotenuse

STEP: Use the definition of sin 69° to get the answer
[−1 point ⇒ 0 / 1 points left]

We need to find the missing quantity in the equation:

sin69°=R

The labels of the sides of a right-angled triangle are: adjacent, opposite, and hypotenuse.

The side that does not touch the 90° angle is called the hypotenuse. In this question, the hypotenuse is the side R. The names of the other two sides depends on the angle of interest. In this question, the angle of interest is the one whose size is 69°. Using this angle:

  • side S is the adjacent side because it is next to (or adjacent to) the 69° angle.
  • side Q is the opposite side because it is across from (or opposite to) the 69° angle.

In order to define the sine of an acute angle in a right-angled triangle, we make use of the opposite and hypotenuse sides. In this case:

sin69°=oppositehypotenuse

In the triangle diagram given, the label of the opposite side is Q and the label for the hypotenuse side is R. We have drawn circles around these labels in the diagram shown below:

NOTE: We have shaded the side S because it is not required in the definition of the sine of the angle 69°.

We can now use these labels to define sine for the angle 69°:

sin69°=QR

Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity Q.

The missing quantity is Q.


Submit your answer as:

Trigonometric ratios

The following diagram shows a right-angled triangle. The sides are labelled with lengths 10, 55, and 15. The two acute angles are labelled α and β. The diagram is not drawn to scale.

Determine the value of sinα in the following triangle.

INSTRUCTION: Give your answer in surd form, if necessary, and simplify your answer completely. Type sqrt( ) if you need to indicate a square root.
Answer: sinα=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Then use the definition of sinα to get the answer.


STEP: Determine the value of sinα for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of sinα.

To find sinα, we need to identify the opposite and the hypotenuse. The opposite side depends on the location of the angle α. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the the opposite side. It is the side across the triangle form the angle α.

In this case, the length of the opposite is 55 and the length of the hypotenuse is equal to 15. Therefore:

sinα=5515=53

Therefore, sinα=53.


Submit your answer as:

Trigonometric ratios

The following diagram shows a right-angled triangle. The sides are labelled with lengths 213, 12, and 14. The two acute angles are labelled θ and λ. The diagram is not drawn to scale.

Determine the value of cosθ in the following triangle.

INSTRUCTION: Give your answer in surd form, if necessary, and simplify your answer completely. Type sqrt( ) if you need to indicate a square root.
Answer: cosθ=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Then use the definition of cosθ to get the answer.


STEP: Determine the value of cosθ for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of cosθ.

To find cosθ, we need to identify the adjacent and the hypotenuse. The adjacent side depends on the location of the angle θ. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the the adjacent side. It is the side next to the angle θ.

In this case, the length of the adjacent is 213 and the length of the hypotenuse is equal to 14. Therefore:

cosθ=21314=137

Therefore, cosθ=137.


Submit your answer as:

Trigonometric ratios

The following diagram shows a right-angled triangle. The sides are labelled with lengths 103, 10, and 20. The two acute angles are labelled A and B. The diagram is not drawn to scale.

Determine the value of tanB in the following triangle.

INSTRUCTION: Give your answer in surd form, if necessary, and simplify your answer completely. Type sqrt( ) if you need to indicate a square root.
Answer: tanB=
expression
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides of the triangle. Then use the definition of tanB to get the answer.


STEP: Determine the value of tanB for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to determine the value of tanB.

To find tanB, we need to identify the opposite and the adjacent. The opposite side is across the triangle from the angle B. The adjacent side is next to B. The arrows and circles below show the those two sides. The opposite side is across the triangle from B, and the adjacent side is next to B.

In this case, the length of the opposite is 10 and the length of the adjacent is equal to 103. Therefore:

tanB=10103=223=33

Therefore, tanB=33.


Submit your answer as:

Trigonometric ratio definitions

The triangle MNP below is right-angled at M.

Use the triangle MNP to answer the following questions:

  1. What is the quantity represented by in the following equation?

    tanN^=MP
    Answer:

    The missing quantity is .

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the angle N^ to identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. After that, use the following definition to answer the question:

    tanN^=oppositeadjacent

    STEP: Use the definition of tanN^ to get the answer
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    tanN^=MP

    The names of the sides of a right-angled triangle are hypotenuse, adjacent, and opposite sides. The side that does not touch the angle M^ is the hypotenuse. In this question, the hypotenuse is the side NP. The names of the other two sides depend on the angle of interest. In this question, the angle of interest is angle N^. Using this angle:

    • side MN is the adjacent side because it is next to (or adjacent to) the angle N^.
    • side MP is the opposite side because it is across from (or opposite to) the angle N^.

    To define tangent we make use of the sides: opposite and adjacent. The definition of tangent for the angle N^ is:

    tanN^=oppositeadjacent

    From the triangle diagram, the label for the opposite side is MP, and the label for the adjacent side is MN.

    Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity MN. We can now complete the equation:

    tanN^=MPMN
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the tangent of the angle N^. These are: MP and MN.

    The hypotenuse side, NP, has been shaded out because it is not needed for the tangent ratio.

    The missing quantity is MN.


    Submit your answer as:
  2. From the drop-down list below, choose the quantity that would make the following equation complete.

    P^=MNNP
    Answer: The missing quantity is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    The missing quantity is a trigonometric function. Using the angle P^, identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. Then use any of the following definitions to answer the question:

    cosP^=adjacenthypotenusesinP^=oppositehypotenusetanP^=oppositeadjacent

    STEP: Identify the trigonometric ratio of the angle P^ equal to the ratio MNNP
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    P^=MNNP

    The missing quantity is a trigonometric function. Trigonometric functions of acute angles of a right-angled triangle are equal to the ratio of two of the sides. The definitions of three trigonometric functions of the angle P^ using the names of the sides, are:

    cosP^=adjacenthypotenusesinP^=oppositehypotenusetanP^=oppositeadjacent

    We now need to identify the names of the sides: MN and NP. Then we will identify the missing trigonometric function.

    For the angle P^, the adjacent is the side MP, the opposite is side MN, and the hypotenuse is side NP. The equation in the question statement has the ratio of the opposite side to the hypotenuse side. Using the equations of the definitions of the three trigonometric ratios above, we find that the missing quantity is sin. The completed equation is:

    sinP^=MNNP
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the sine of the angle P^. These are: the opposite side MN and the hypotenuse side NP.

    The missing quantity is sin.


    Submit your answer as:

Trigonometric ratio definitions

The triangle PQR below is right-angled at P.

Use the triangle PQR to answer the following questions:

  1. What is the quantity represented by in the following equation?

    sinR^=PQ
    Answer:

    The missing quantity is .

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the angle R^ to identify the hypotenuse, adjacent, and opposite sides of the triangle PQR. After that, use the following definition to answer the question:

    sinR^=oppositehypotenuse

    STEP: Use the definition of sinR^ to get the answer
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    sinR^=PQ

    The names of the sides of a right-angled triangle are hypotenuse, adjacent, and opposite sides. The side that does not touch the angle P^ is the hypotenuse. In this question, the hypotenuse is the side QR. The names of the other two sides depend on the angle of interest. In this question, the angle of interest is angle R^. Using this angle:

    • side PR is the adjacent side because it is next to (or adjacent to) the angle R^.
    • side PQ is the opposite side because it is across from (or opposite to) the angle R^.

    To define sine we make use of the sides: opposite and hypotenuse. The definition of sine for the angle R^ is:

    sinR^=oppositehypotenuse

    From the triangle diagram, the label for the opposite side is PQ, and the label for the hypotenuse side is QR.

    Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity QR. We can now complete the equation:

    sinR^=PQQR
    NOTE:

    We will redraw the triangle PQR to show the two sides we need for the sine of the angle R^. These are: PQ and QR.

    The adjacent side, PR, has been shaded out because it is not needed for the sine ratio.

    The missing quantity is QR.


    Submit your answer as:
  2. From the drop-down list below, choose the quantity that would make the following equation complete.

    R^=PQPR
    Answer: The missing quantity is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    The missing quantity is a trigonometric function. Using the angle R^, identify the hypotenuse, adjacent, and opposite sides of the triangle PQR. Then use any of the following definitions to answer the question:

    cosR^=adjacenthypotenusesinR^=oppositehypotenusetanR^=oppositeadjacent

    STEP: Identify the trigonometric ratio of the angle R^ equal to the ratio PQPR
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    R^=PQPR

    The missing quantity is a trigonometric function. Trigonometric functions of acute angles of a right-angled triangle are equal to the ratio of two of the sides. The definitions of three trigonometric functions of the angle R^ using the names of the sides, are:

    cosR^=adjacenthypotenusesinR^=oppositehypotenusetanR^=oppositeadjacent

    We now need to identify the names of the sides: PQ and PR. Then we will identify the missing trigonometric function.

    For the angle R^, the adjacent is the side PR, the opposite is side PQ, and the hypotenuse is side QR. The equation in the question statement has the ratio of the opposite side to the adjacent side. Using the equations of the definitions of the three trigonometric ratios above, we find that the missing quantity is tan. The completed equation is:

    tanR^=PQPR
    NOTE:

    We will redraw the triangle PQR to show the two sides we need for the tangent of the angle R^. These are: the opposite side PQ and the adjacent side PR.

    The missing quantity is tan.


    Submit your answer as:

Trigonometric ratio definitions

The triangle MNP below is right-angled at M.

Use the triangle MNP to answer the following questions:

  1. What is the quantity represented by in the following equation?

    cosN^=MN
    Answer:

    The missing quantity is .

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the angle N^ to identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. After that, use the following definition to answer the question:

    cosN^=adjacenthypotenuse

    STEP: Use the definition of cosN^ to get the answer
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    cosN^=MN

    The names of the sides of a right-angled triangle are hypotenuse, adjacent, and opposite sides. The side that does not touch the angle M^ is the hypotenuse. In this question, the hypotenuse is the side NP. The names of the other two sides depend on the angle of interest. In this question, the angle of interest is angle N^. Using this angle:

    • side MN is the adjacent side because it is next to (or adjacent to) the angle N^.
    • side MP is the opposite side because it is across from (or opposite to) the angle N^.

    To define cosine we make use of the sides: adjacent and hypotenuse. The definition of cosine for the angle N^ is:

    cosN^=adjacenthypotenuse

    From the triangle diagram, the label for the adjacent side is MN, and the label for the hypotenuse side is NP.

    Comparing this to the equation with the unknown quantity, we can see that the identity of is the quantity NP. We can now complete the equation:

    cosN^=MNNP
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the cosine of the angle N^. These are: MN and NP.

    The opposite side, MP, has been shaded out because it is not needed for the cosine ratio.

    The missing quantity is NP.


    Submit your answer as:
  2. From the drop-down list below, choose the quantity that would make the following equation complete.

    N^=MPNP
    Answer: The missing quantity is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    The missing quantity is a trigonometric function. Using the angle N^, identify the hypotenuse, adjacent, and opposite sides of the triangle MNP. Then use any of the following definitions to answer the question:

    cosN^=adjacenthypotenusesinN^=oppositehypotenusetanN^=oppositeadjacent

    STEP: Identify the trigonometric ratio of the angle N^ equal to the ratio MPNP
    [−1 point ⇒ 0 / 1 points left]

    We need to find the missing quantity in the equation:

    N^=MPNP

    The missing quantity is a trigonometric function. Trigonometric functions of acute angles of a right-angled triangle are equal to the ratio of two of the sides. The definitions of three trigonometric functions of the angle N^ using the names of the sides, are:

    cosN^=adjacenthypotenusesinN^=oppositehypotenusetanN^=oppositeadjacent

    We now need to identify the names of the sides: MP and NP. Then we will identify the missing trigonometric function.

    For the angle N^, the adjacent is the side MN, the opposite is side MP, and the hypotenuse is side NP. The equation in the question statement has the ratio of the opposite side to the hypotenuse side. Using the equations of the definitions of the three trigonometric ratios above, we find that the missing quantity is sin. The completed equation is:

    sinN^=MPNP
    NOTE:

    We will redraw the triangle MNP to show the two sides we need for the sine of the angle N^. These are: the opposite side MP and the hypotenuse side NP.

    The missing quantity is sin.


    Submit your answer as:

Evaluating trigonometric ratios

The diagram below shows a right-angled triangle with sides of length 40, 30, and 50. The two acute angles are labelled θ and λ.

Determine the value of tanθ for this triangle.

INSTRUCTION: Give your answer as a simplified fraction.
Answer: tanθ=
fraction
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Start by writing down the definition of the tan ratio. Then identify which sides of the triangle correspond to the sides in the definition to find the answer.


STEP: Determine the value of tanθ for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio tanθ.

The definition of tanθ is

tanθ=oppositeadjacent

So we need to identify the opposite and adjacent sides for the angle θ. The opposite side is across the triangle from the angle θ. The adjacent side is next to θ. The arrows and circles below show those two sides. The opposite side is across the triangle from θ, and the adjacent side is next to θ.

In this case, the length of the opposite is 30 and the length of the adjacent is 40. Therefore:

tanθ=oppositeadjacent=3040=34

Therefore, the answer is tanθ=34.


Submit your answer as:

Evaluating trigonometric ratios

The diagram below shows a right-angled triangle with sides of length 16, 34, and 30. The two acute angles are labelled α and β.

Determine the value of sinα for this triangle.

INSTRUCTION: Give your answer as a simplified fraction.
Answer: sinα=
fraction
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Start by writing down the definition of the sin ratio. Then identify which sides of the triangle correspond to the sides in the definition to find the answer.


STEP: Determine the value of sinα for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio sinα.

The definition of sinα is

sinα=oppositehypotenuse

So we need to identify the opposite and hypotenuse sides for the angle α. The opposite side depends on the location of the angle α. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the opposite side. It is the side across the triangle from the angle α.

In this case, the length of the opposite is 30 and the length of the hypotenuse is 34. Therefore:

sinα=oppositehypotenuse=3034=1517

Therefore, the answer is sinα=1517.


Submit your answer as:

Evaluating trigonometric ratios

The diagram below shows a right-angled triangle with sides of length 24, 18, and 30. The two acute angles are labelled A and B.

Determine the value of sinA for this triangle.

INSTRUCTION: Give your answer as a simplified fraction.
Answer: sinA=
fraction
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Start by writing down the definition of the sin ratio. Then identify which sides of the triangle correspond to the sides in the definition to find the answer.


STEP: Determine the value of sinA for the given triangle
[−2 points ⇒ 0 / 2 points left]

We have been given the lengths of the three sides of a right-angled triangle. We need to evaluate the ratio sinA.

The definition of sinA is

sinA=oppositehypotenuse

So we need to identify the opposite and hypotenuse sides for the angle A. The opposite side depends on the location of the angle A. (The hypotenuse is always the side across from the right-angle.) The arrow and circle below show the opposite side. It is the side across the triangle from the angle A.

In this case, the length of the opposite is 18 and the length of the hypotenuse is 30. Therefore:

sinA=oppositehypotenuse=1830=35

Therefore, the answer is sinA=35.


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2. Practical applications

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is 7 meters, Point A is x meters away from the foot of the building, and the angle of incline (from the horizontal) the the top of the building is 65°.

Calculate the distance away from the foot of the building (x) as shown in the diagram below. Round your answer to two decimal places where necessary.

Answer: x = m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the tangent trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the correct trig ratio to use is the tangent function.

tan(θ)=oppositeadjacenttan(65)°=7xx=7tan(65)°x=3,2641...x3,26

Remember to round your answer to two decimal places. Therefore x is 3,26 m.


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Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, at Point B.

The height of the building is 5 m, the line-of-sight distance from Point A to the top of the building (Point B) is x, and the angle of inclination (from the horizontal) to the top of the building is 61°.

Calculate the line-of-sight distance to the top of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The line-of-sight distance to the top of the building is m.
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the sine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is sin.

sinθ=oppositehypotenusesin61°=5xx=5sin61°x=5,71677...5,72 m

The line-of-sight distance to the top of the building is 5,72 m.


Submit your answer as:

Using trigonometry with word problems

A man stands at a window on the first floor of a building, shouting down to a child on the other side of the street. The following are true:

  1. The man in the window is 3 m above the street.
  2. The horizontal distance from the man standing in the window to the point directly above the child on the street is 4,5 m.

Find the angle of depression from the man down to the child.

INSTRUCTION: Round your answer to two decimal places.
Answer: The angle is °.
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Start by drawing a triangle diagram to represent the situation in the question. Label all of the information given in the question. Also label any other information you know is true (for example, if the triangle is equilateral, mark all three sides to show they are the same length).


STEP: Draw a diagram to represent the situation and label it
[−1 point ⇒ 3 / 4 points left]

When you have a word problem about a shape or positions and there is no diagram, start by drawing a sketch to summarise the question. Label the sketch with all of the information given. In this case, the information given in the question fits on a triangle: there is one thing at ground level (the child on the street) and something else up above (the man in the window). We can represent this with a right-angled triangle.

The picture does not need to be perfect but try to make it reasonable. For example, the question gives you two lengths, one of which is longer than the other: draw the picture so that the longer side is clearly longer than the shorter length.

NOTE: The diagram should also have a label for what you want to find. We have used "a" to label the angle that we want to find (but you can use any variable that you want).

STEP: Determine which trigonometric ratio to use
[−1 point ⇒ 2 / 4 points left]

One of the most important facts about the picture above is that it is a right-angled triangle because the street and the building are perpendicular to each other. That means we can use the trigonometric ratios to try to solve this question, or even the theorem of Pythagoras: we should pick whichever option will get us the answer.

In this case, the sides given are the two sides opposite from and adjacent to the angle we want. Therefore, the correct trigonometric ratio is the tangent function. (In fact, there is no other way to solve the question with the information given. While the theorem of Pythagoras applies to the triangle, it is about the lengths of the sides, and we need to find an angle: the theorem is useless for this question.)


STEP: Set up an equation and solve for the answer
[−2 points ⇒ 0 / 4 points left]

Now we can write an equation for the tangent ratio from the sketch, and solve for the angle we need. Start by writing the equation for tangent, and then substitute in the correct values.

tanθ=oppositeadjacenttana=34,5a=tan1(34,5)a=33,6900...a33,69°

We got it! As shown in the last step, we rounded off to the nearest hundredths place because the question told us to.

The angle of depression from the man down to the child is 33,69°.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a tree, Point B.

Point A is 9.0 meters away from the foot of the tree, the line-of-sight distance from Point A to the top of the tree (Point B) is x meters, and the angle of incline (from the horizontal) the the top of the tree is 45°.

Calculate the line-of-sight distance to the top of the tree (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: x= m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the cosine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is cos.

cosθ=adjacenthypotenusecos45°=9.0xx=12,73 m

Therefore x is 12,73 m.


Submit your answer as:

Trigonometry with non-right triangles

The figure below, which is drawn to scale, shows an isosceles triangle. Side AC¯=7,3 is labelled and A^=62,7°. Point D is on side AB¯ directly across from point C, such that CD¯ makes a right angle with AB¯.

  1. Which of the following statements must be true about the figure above?

    Answer: The true statement is: .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Remember that isosceles triangles are symmetric.


    STEP: Consider the choices and select the correct option
    [−1 point ⇒ 0 / 1 points left]

    The question asks us to select the true statement from the choices in the list. Note that two of the options must be wrong straightaway: ΔABC is neither a right-angled triangle nor equilateral. From the remaining options, the correct choice is based on the fact that the segment CD¯ divides ΔABC into two identical shapes: the segment CD¯ is a line of symmetry because the triangle is isosceles. Therefore, point D divides AB¯ into two equal parts.

    The correct choice from the list is: AD¯ = DB¯.


    Submit your answer as:
  2. Compute the length of AB¯. Round your answer to one decimal place.

    Answer: The length of AB¯ is .
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the segment CD¯ to get two right-angled triangles. Then you can use trigonometric ratios or the theorem of Pythagoras to work out the answer to the question.


    STEP: Draw a line to create right-angled triangles
    [−1 point ⇒ 2 / 3 points left]

    The first thing to do is draw an extra line across the triangle so that we make two right-angled triangles in the figure. We do this because we can use the trigonometric ratios and the theorem of Pythagoras for right-angled triangles.

    The line segment DC¯ in the figure is the line we want: it will create two separate right-angled triangles! The two right-angled triangles that we get look like this:


    STEP: Use trigonometry to find useful information
    [−1 point ⇒ 1 / 3 points left]

    In ΔADC (the light blue one) we know one of the non-right angles and one of the sides. Hence we can use the trigonometric ratios in that triangle because it is a right-angled triangle.

    With the information given, we can find both segments AD¯ and CD¯. However, for this question only AD¯ is useful: we want the length of AB¯, which is twice as long as AD¯. In ΔADC, the hypotenuse is AC¯=7,3, and the side we want is adjacent to the angle given. Therefore we need to use the cosine ratio. Set up the equation and then solve for the length of AD¯.

    cosθ=adjacenthypotenusecos(62,7°)=AD¯7,3(7,3)cos62,7°=AD¯(7,3)(0,4586...)=AD¯3,3481...=AD¯


    STEP: Calculate the final answer
    [−1 point ⇒ 0 / 3 points left]

    We want the length of AB¯, so multiply by two since D is the mid-point of segment AB¯:

    AB¯=2AD¯=2(3,3481...)=6,6962...6,7

    Remember that the instructions say to round the answer to the first decimal place, as shown in the last step above.

    The final answer is: AB¯=6,7.


    Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is 6 m, Point A is x  away from the foot of the building, and the line-of-sight distance from Point A to the top of the building is 7,32 m.

Calculate the distance away from the foot of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The distance away from the foot of the building is m.
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Use the Theorem of Pythagoras to calculate the distance away from the foot of the building.


STEP: Use the Theorem of Pythagoras to calculate the distance away from the foot of the building
[−2 points ⇒ 0 / 2 points left]

The question gives us two sides of a right-angled triangle, and we need to find the third side: we can use the Theorem of Pythagoras.

c2=a2+b2(7,32)2=(6)2+x2Solving: x=4,19313...4,2 m

Therefore, the distance away from the foot of the building is 4,2 m.


Submit your answer as:

Exercises

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is x meters, Point A is 7.0 meters away from the foot of the building, and the angle of incline (from the horizontal) the the top of the building is 55°.

Calculate the height of the building (x) as shown in the diagram below. Round your answer to two decimal places where necessary.

Answer: x = m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the tangent trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the correct trig ratio to use is the tangent function.

tan(θ)=oppositeadjacenttan(55)°=x7,0x=7,0tan(55)°x=9,9970...x10

Remember to round your answer to two decimal places. Therefore x is 10 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is x meters, Point A is 5.6 meters away from the foot of the building, and the angle of incline (from the horizontal) the the top of the building is 55°.

Calculate the height of the building (x) as shown in the diagram below. Round your answer to two decimal places where necessary.

Answer: x = m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the tangent trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the correct trig ratio to use is the tangent function.

tan(θ)=oppositeadjacenttan(55)°=x5,6x=5,6tan(55)°x=7,9976...x8

Remember to round your answer to two decimal places. Therefore x is 8 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is x meters, Point A is 4.04 meters away from the foot of the building, and the angle of incline (from the horizontal) the the top of the building is 60°.

Calculate the height of the building (x) as shown in the diagram below. Round your answer to two decimal places where necessary.

Answer: x = m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the tangent trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the correct trig ratio to use is the tangent function.

tan(θ)=oppositeadjacenttan(60)°=x4,04x=4,04tan(60)°x=6,9974...x7

Remember to round your answer to two decimal places. Therefore x is 7 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, at Point B.

The height of the building is 9 m, the line-of-sight distance from Point A to the top of the building (Point B) is x, and the angle of inclination (from the horizontal) to the top of the building is 52°.

Calculate the line-of-sight distance to the top of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The line-of-sight distance to the top of the building is m.
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the sine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is sin.

sinθ=oppositehypotenusesin52°=9xx=9sin52°x=11,42116...11,42 m

The line-of-sight distance to the top of the building is 11,42 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, at Point B.

The height of the building is 5 m, the line-of-sight distance from Point A to the top of the building (Point B) is 5,72 m, and the angle of inclination (from the horizontal) to the top of the building is x.

Calculate the angle of inclination (from the horizontal) to the top of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The angle of inclination (from the horizontal) to the top of the building is °.
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the sine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is sin.

sinθ=oppositehypotenusesinx=55,72x=sin1(55,72)x=60,94169...60,94 m

The angle of inclination (from the horizontal) to the top of the building is 60,94 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a tree, at Point B.

The height of the tree is 10 m, the line-of-sight distance from Point A to the top of the tree (Point B) is 11,03 m, and the angle of inclination (from the horizontal) to the top of the tree is x.

Calculate the angle of inclination (from the horizontal) to the top of the tree (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The angle of inclination (from the horizontal) to the top of the tree is °.
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the sine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is sin.

sinθ=oppositehypotenusesinx=1011,03x=sin1(1011,03)x=65,04213...65,04 m

The angle of inclination (from the horizontal) to the top of the tree is 65,04 m.


Submit your answer as:

Using trigonometry with word problems

At a house in Polokwane, a father stands at the top of the stairs. His child is at the bottom of the stairs. The following are true:

  1. At the bottom of the stairs, there is an angle of 46,4° between the slope of the stairs and the vertical direction.
  2. The horizontal distance from the father at the top of the stairs to the point straight up from the child is 4,2 m.

Calculate the vertical distance of the father above the child (the height from the top of the stairs to the bottom).

INSTRUCTION: Round your answer to two decimal places.
Answer: The distance is metres.
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Start by drawing a triangle diagram to represent the situation in the question. Label all of the information given in the question. Also label any other information you know is true (for example, if the triangle is equilateral, mark all three sides to show they are the same length).


STEP: Draw a diagram to represent the situation and label it
[−1 point ⇒ 3 / 4 points left]

When you have a word problem about a shape or positions and there is no diagram, start by drawing a sketch to summarise the question. Label the sketch with all of the information given. In this case, the information given in the question fits on a triangle: there is one thing at ground level (the child at the bottom of the stairs) and something else up above (the father at the top of the stairs). We can represent this with a right-angled triangle.

The picture does not need to be perfect but try to make it reasonable. For example, this question gives an angle of 46,4°: try to make the angle in the sketch reasonably close to that size.

NOTE: The diagram should also have a label for what you want to find. We have used "y" to label the distance that we want to find (but you can use any variable that you want).

STEP: Determine which trigonometric ratio to use
[−1 point ⇒ 2 / 4 points left]

One of the most important facts about the picture above is that it is a right-angled triangle: we can draw a vertical line straight up from the child, which is perpendicular to the horizontal line shown. That means we can use the trigonometric ratios to try to solve this question, or even the theorem of Pythagoras: we should pick whichever option will get us the answer.

In this case, we do not know anything about the hypotenuse. Instead, with respect to the angle given, we know the opposite side and we need to determine the length of the adjacent side. Therefore, the correct trigonometric ratio is the tangent function. (In fact, there is no other way to solve the question with the information given. While the theorem of Pythagoras applies to the triangle, we do not know enough information to use it for this question.)


STEP: Set up an equation and solve for the answer
[−2 points ⇒ 0 / 4 points left]

Now we can write an equation for the tangent ratio from the sketch, and solve for the distance we need. Start by writing the equation for tangent, and then substitute in the correct values.

tanθ=oppositeadjacenttan(46,4°)=4,2yy×tan46,4°=4,2y×yto clear the denominatorMultiply both sides by yy=4,2tan46,4°y=3,9996...y4 m

We got it! As shown in the last step, we rounded off to the nearest hundredths place because the question told us to. (In this case, rounding off to two decimal places results in an integer value. If you give the answer as 4,00 that is fine.)

The vertical distance of the father above the child (the height from the top of the stairs to the bottom) is 4 metres.


Submit your answer as:

Using trigonometry with word problems

A man stands at a window on the first floor of a building, shouting down to a child on the other side of the street. Here are two facts about the situation:

  1. For the child on the street, the line to the man in the window makes an angle of 60° with the vertical direction.
  2. The man in the window is 3 m above the street.

Find the oblique (slanted) distance between the man and the child.

INSTRUCTION: Round your answer to two decimal places.
Answer: The distance is metres.
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Start by drawing a triangle diagram to represent the situation in the question. Label all of the information given in the question. Also label any other information you know is true (for example, if the triangle is equilateral, mark all three sides to show they are the same length).


STEP: Draw a diagram to represent the situation and label it
[−1 point ⇒ 3 / 4 points left]

When you have a word problem about a shape or positions and there is no diagram, start by drawing a sketch to summarise the question. Label the sketch with all of the information given. In this case, the information given in the question fits on a triangle: there is one thing at ground level (the child on the street) and something else up above (the man in the window). We can represent this with a right-angled triangle.

The picture does not need to be perfect but try to make it reasonable. For example, this question gives an angle of 60°: try to make the angle in the sketch reasonably close to that size.

NOTE: The diagram should also have a label for what you want to find. We have used "h" to label the distance that we want to find (but you can use any variable that you want).

STEP: Determine which trigonometric ratio to use
[−1 point ⇒ 2 / 4 points left]

One of the most important facts about the picture above is that it is a right-angled triangle because the street and the building are perpendicular to each other. That means we can use the trigonometric ratios to try to solve this question, or even the theorem of Pythagoras: we should pick whichever option will get us the answer.

In this question we need to find the hypotenuse of the triangle, and the side that we already know is adjacent to the angle given. Therefore, the correct trigonometric ratio is the cosine function. (In fact, there is no other way to solve the question with the information given. While the theorem of Pythagoras applies to the triangle, we do not know enough information to use it for this question.)


STEP: Set up an equation and solve for the answer
[−2 points ⇒ 0 / 4 points left]

Now we can write an equation for the cosine ratio from the sketch, and solve for the distance we need. Start by writing the equation for cosine, and then substitute in the correct values.

cosθ=adjacenthypotenusecos(60°)=3hh×cos60°=3h×hto clear the denominatorMultiply both sides by hh=3cos60°h=6h6 m

We got it! As shown in the last step, we rounded off to the nearest hundredths place because the question told us to. (In this case, rounding off to two decimal places results in an integer value. If you give the answer as 6,00 that is fine.)

The oblique (slanted) distance between the man and the child is 6 metres.


Submit your answer as:

Using trigonometry with word problems

Imagine a cat sitting up in a tree watching a mouse on the ground. Consider these two statements about the situation:

  1. The distance of the cat above the ground is 7 m.
  2. The angle at the cat between the mouse and the vertical direction is 16,6°.

Determine the oblique (slanted) distance between the mouse and the cat.

INSTRUCTION: Round your answer to two decimal places.
Answer: The distance is metres.
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Start by drawing a triangle diagram to represent the situation in the question. Label all of the information given in the question. Also label any other information you know is true (for example, if the triangle is equilateral, mark all three sides to show they are the same length).


STEP: Draw a diagram to represent the situation and label it
[−1 point ⇒ 3 / 4 points left]

When you have a word problem about a shape or positions and there is no diagram, start by drawing a sketch to summarise the question. Label the sketch with all of the information given. In this case, the information given in the question fits on a triangle: there is one thing at ground level (the mouse on the ground) and something else up above (the cat in the tree). We can represent this with a right-angled triangle.

The picture does not need to be perfect but try to make it reasonable. For example, this question gives an angle of 16,6°: try to make the angle in the sketch reasonably close to that size.

NOTE: The diagram should also have a label for what you want to find. We have used "h" to label the distance that we want to find (but you can use any variable that you want).

STEP: Determine which trigonometric ratio to use
[−1 point ⇒ 2 / 4 points left]

One of the most important facts about the picture above is that it is a right-angled triangle: we can draw a vertical line straight down from the cat, which is perpendicular to the ground. That means we can use the trigonometric ratios to try to solve this question, or even the theorem of Pythagoras: we should pick whichever option will get us the answer.

In this question we need to find the hypotenuse of the triangle, and the side that we already know is adjacent to the angle given. Therefore, the correct trigonometric ratio is the cosine function. (In fact, there is no other way to solve the question with the information given. While the theorem of Pythagoras applies to the triangle, we do not know enough information to use it for this question.)


STEP: Set up an equation and solve for the answer
[−2 points ⇒ 0 / 4 points left]

Now we can write an equation for the cosine ratio from the sketch, and solve for the distance we need. Start by writing the equation for cosine, and then substitute in the correct values.

cosθ=adjacenthypotenusecos(16,6°)=7hh×cos16,6°=7h×hto clear the denominatorMultiply both sides by hh=7cos16,6°h=7,3044...h7,3 m

We got it! As shown in the last step, we rounded off to the nearest hundredths place because the question told us to. (In this case, rounding off to two decimal places leads to a number which stops after the first decimal place. However, you can give your answer as 7,30 if you want to.)

The oblique (slanted) distance between the mouse and the cat is 7,3 metres.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a tree, Point B.

Point A is 7.0 meters away from the foot of the tree, the line-of-sight distance from Point A to the top of the tree (Point B) is 9,9 meters, and the angle of incline (from the horizontal) the the top of the tree is x°.

Calculate the angle of incline (from the horizontal) the the top of the tree (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the cosine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is cos.

cosθ=adjacenthypotenusecosx°=7.09,9x=45°

Therefore x is 45°.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a tree, Point B.

Point A is 7.0 meters away from the foot of the tree, the line-of-sight distance from Point A to the top of the tree (Point B) is 9,9 meters, and the angle of incline (from the horizontal) the the top of the tree is x°.

Calculate the angle of incline (from the horizontal) the the top of the tree (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the cosine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is cos.

cosθ=adjacenthypotenusecosx°=7.09,9x=45°

Therefore x is 45°.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

Point A is 4.2 meters away from the foot of the building, the line-of-sight distance from Point A to the top of the building (Point B) is x meters, and the angle of incline (from the horizontal) the the top of the building is 55°.

Calculate the line-of-sight distance to the top of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: x= m
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Identify the opposite, adjacent and hypotenuse sides and use the appropriate trigonometric ratio to solve for x.


STEP: Use the cosine trigonometric ratio to solve for x
[−2 points ⇒ 0 / 2 points left]

First identify the opposite, adjacent and hypotenuse sides. Then use the appropriate trigonometric ratio to solve for x. In this example, the trig ratio to use is cos.

cosθ=adjacenthypotenusecos55°=4.2xx=7,32 m

Therefore x is 7,32 m.


Submit your answer as:

Trigonometry with non-right triangles

The figure below, which is drawn to scale, shows a scalene triangle. Two sides and an angle are given: DF¯=8,9, DE¯=9 and D^=52,4°. Point G is on side DE¯ as labelled, such that FG¯ makes a right angle with DE¯.

  1. Which of the following statements must be true about the figure above?

    Answer: The true statement is: .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Imagine walking from point D to G and then walking on from G to E. How does this compare to walking straight from D to E without stopping?


    STEP: Consider the choices and select the correct option
    [−1 point ⇒ 0 / 1 points left]

    The question asks us to select the true statement from the choices in the list. The correct option is about the sum of the segments along side DE¯. Point G breaks DE¯ up into two pieces. Therefore we know that together they make the total length of DE¯. Note that we do not know the exact position of point G, only that it is somewhere between D and E.

    The correct choice from the list is: Side DE is the sum of sides DG and GE.


    Submit your answer as:
  2. Now determine the length of EF¯. Round your answer to one decimal place.

    Answer: The length of EF¯ is .
    numeric
    HINT: <no title>
    [−0 points ⇒ 7 / 7 points left]

    Use the segment FG¯ to get two right-angled triangles. Then you can use trigonometric ratios or the theorem of Pythagoras to work out the answer to the question.


    STEP: Draw a line to create right-angled triangles
    [−1 point ⇒ 6 / 7 points left]

    The first thing to do is draw an extra line across the triangle so that we make two right-angled triangles in the figure. We do this because we can use the trigonometric ratios and the theorem of Pythagoras for right-angled triangles.

    The line segment GF¯ in the figure is the line we want: it will create two separate right-angled triangles! The two right-angled triangles that we get look like this:


    STEP: Use trigonometry to find useful information
    [−4 points ⇒ 2 / 7 points left]

    In ΔDGF (the light blue one) we know one of the non-right angles and one of the sides. Hence we can use the trigonometric ratios in that triangle because it is a right-angled triangle.

    With the information given, we can find both segments DG¯ and FG¯. Remember that we want to get the length of EF¯, and for that we need both of these lengths. Start by calculating the length of DG¯, which will allow us to find GE¯ (because we know that DE¯=9). This calculation involves the hypotenuse and the side adjacent to D^, so use the cosine ratio.

    cosθ=adjacenthypotenusecos(52,4°)=DG¯8,9(8,9)cos52,4°=DG¯(8,9)(0,6101...)=DG¯5,4302...=DG¯SinceDG¯+GE¯=DE¯:GE¯=95,4302...GE¯=3,5697...

    Great: that gets us the value for side GE¯. Now we need to find the length of side FG¯. For that we will use the sine ratio (you can also do this calculation with the theorem of Pythagoras, but here we will do it with trigonometry).

    sinθ=oppositehypotenusesin(52,4°)=FG¯8,9(8,9)sin52,4°=FG¯(8,9)(0,7922...)=FG¯7,0513...=FG¯

    STEP: Calculate the final answer
    [−2 points ⇒ 0 / 7 points left]

    Now we have the lengths of two sides of ΔGEF; since ΔGEF is a right-angled triangle, we can use the theorem of Pythagoras to calculate the length of the hypotenuse.

    c2=a2+b2(EF¯)2=(FG¯)2+(GE¯)2=(7,0513...)2+(3,5697...)2=62,4647...EF¯=±62,4647...the ± because a2=(a)2The square root brings in=±7,9034...±7,9

    Remember that the instructions say to round the answer to the first decimal place, as shown in the last step above. Also notice that we get two different answers, one positive and one negative. However, the value we calculated represents a distance, so we must throw out the negative answer.

    The final answer is: EF¯=7,9.


    Submit your answer as:

Trigonometry with non-right triangles

The figure below, which is drawn to scale, shows a scalene triangle. Two sides and an angle are given: PR¯=5, PQ¯=6,1 and P^=38,5°. Point S is on side PQ¯ as labelled, such that RS¯ makes a right angle with PQ¯.

  1. Which of the following statements must be true about the figure above?

    Answer: The true statement is: .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Imagine walking from point P to S and then walking on from S to Q. How does this compare to walking straight from P to Q without stopping?


    STEP: Consider the choices and select the correct option
    [−1 point ⇒ 0 / 1 points left]

    The question asks us to select the true statement from the choices in the list. The correct option is about the sum of the segments along side PQ¯. Point S breaks PQ¯ up into two pieces. Therefore we know that together they make the total length of PQ¯. Note that we do not know the exact position of point S, only that it is somewhere between P and Q.

    The correct choice from the list is: Side PQ is the sum of sides PS and SQ.


    Submit your answer as:
  2. Now determine the length of QR¯. Round your answer to one decimal place.

    Answer: The length of QR¯ is .
    numeric
    HINT: <no title>
    [−0 points ⇒ 7 / 7 points left]

    Use the segment RS¯ to get two right-angled triangles. Then you can use trigonometric ratios or the theorem of Pythagoras to work out the answer to the question.


    STEP: Draw a line to create right-angled triangles
    [−1 point ⇒ 6 / 7 points left]

    The first thing to do is draw an extra line across the triangle so that we make two right-angled triangles in the figure. We do this because we can use the trigonometric ratios and the theorem of Pythagoras for right-angled triangles.

    The line segment SR¯ in the figure is the line we want: it will create two separate right-angled triangles! The two right-angled triangles that we get look like this:


    STEP: Use trigonometry to find useful information
    [−4 points ⇒ 2 / 7 points left]

    In ΔPSR (the light blue one) we know one of the non-right angles and one of the sides. Hence we can use the trigonometric ratios in that triangle because it is a right-angled triangle.

    With the information given, we can find both segments PS¯ and RS¯. Remember that we want to get the length of QR¯, and for that we need both of these lengths. Start by calculating the length of PS¯, which will allow us to find SQ¯ (because we know that PQ¯=6,1). This calculation involves the hypotenuse and the side adjacent to P^, so use the cosine ratio.

    cosθ=adjacenthypotenusecos(38,5°)=PS¯5(5)cos38,5°=PS¯(5)(0,7826...)=PS¯3,9130...=PS¯SincePS¯+SQ¯=PQ¯:SQ¯=6,13,9130...SQ¯=2,1869...

    Great: that gets us the value for side SQ¯. Now we need to find the length of side RS¯. For that we will use the sine ratio (you can also do this calculation with the theorem of Pythagoras, but here we will do it with trigonometry).

    sinθ=oppositehypotenusesin(38,5°)=RS¯5(5)sin38,5°=RS¯(5)(0,6225...)=RS¯3,1125...=RS¯

    STEP: Calculate the final answer
    [−2 points ⇒ 0 / 7 points left]

    Now we have the lengths of two sides of ΔSQR; since ΔSQR is a right-angled triangle, we can use the theorem of Pythagoras to calculate the length of the hypotenuse.

    c2=a2+b2(QR¯)2=(RS¯)2+(SQ¯)2=(3,1125...)2+(2,1869...)2=14,4709...QR¯=±14,4709...the ± because a2=(a)2The square root brings in=±3,8040...±3,8

    Remember that the instructions say to round the answer to the first decimal place, as shown in the last step above. Also notice that we get two different answers, one positive and one negative. However, the value we calculated represents a distance, so we must throw out the negative answer.

    The final answer is: QR¯=3,8.


    Submit your answer as:

Trigonometry with non-right triangles

The figure below, which is drawn to scale, shows an isosceles triangle. Side VX¯=7,1 is labelled and V^=58,6°. Point Y is on side VW¯ directly across from point X, such that XY¯ makes a right angle with VW¯.

  1. Which of the following statements must be true about the figure above?

    Answer: The true statement is: .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Remember that isosceles triangles are symmetric.


    STEP: Consider the choices and select the correct option
    [−1 point ⇒ 0 / 1 points left]

    The question asks us to select the true statement from the choices in the list. Note that two of the options must be wrong straightaway: ΔVWX is neither a right-angled triangle nor equilateral. From the remaining options, the correct choice is based on the fact that the segment XY¯ divides ΔVWX into two identical shapes: the segment XY¯ is a line of symmetry because the triangle is isosceles. Therefore, point Y divides VW¯ into two equal parts.

    The correct choice from the list is: Point Y divides side VW into two equal pieces.


    Submit your answer as:
  2. Next, find the length of VW¯. Round your answer to one decimal place.

    Answer: The length of VW¯ is .
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the segment XY¯ to get two right-angled triangles. Then you can use trigonometric ratios or the theorem of Pythagoras to work out the answer to the question.


    STEP: Draw a line to create right-angled triangles
    [−1 point ⇒ 2 / 3 points left]

    The first thing to do is draw an extra line across the triangle so that we make two right-angled triangles in the figure. We do this because we can use the trigonometric ratios and the theorem of Pythagoras for right-angled triangles.

    The line segment YX¯ in the figure is the line we want: it will create two separate right-angled triangles! The two right-angled triangles that we get look like this:


    STEP: Use trigonometry to find useful information
    [−1 point ⇒ 1 / 3 points left]

    In ΔVYX (the light blue one) we know one of the non-right angles and one of the sides. Hence we can use the trigonometric ratios in that triangle because it is a right-angled triangle.

    With the information given, we can find both segments VY¯ and XY¯. However, for this question only VY¯ is useful: we want the length of VW¯, which is twice as long as VY¯. In ΔVYX, the hypotenuse is VX¯=7,1, and the side we want is adjacent to the angle given. Therefore we need to use the cosine ratio. Set up the equation and then solve for the length of VY¯.

    cosθ=adjacenthypotenusecos(58,6°)=VY¯7,1(7,1)cos58,6°=VY¯(7,1)(0,5210...)=VY¯3,6991...=VY¯


    STEP: Calculate the final answer
    [−1 point ⇒ 0 / 3 points left]

    We want the length of VW¯, so multiply by two since Y is the mid-point of segment VW¯:

    VW¯=2VY¯=2(3,6991...)=7,3983...7,4

    Remember that the instructions say to round the answer to the first decimal place, as shown in the last step above.

    The final answer is: VW¯=7,4.


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Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is 7 m, Point A is 8,34 m away from the foot of the building, and the line-of-sight distance from Point A to the top of the building is x .

Calculate the line-of-sight distance to the top of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The line-of-sight distance to the top of the building is m.
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Use the Theorem of Pythagoras to calculate the line-of-sight distance to the top of the building.


STEP: Use the Theorem of Pythagoras to calculate the line-of-sight distance to the top of the building
[−2 points ⇒ 0 / 2 points left]

The question gives us two sides of a right-angled triangle, and we need to find the third side: we can use the Theorem of Pythagoras.

c2=a2+b2x2=(7)2+(8,34)2Solving: x=10,88832...10,89 m

Therefore, the line-of-sight distance to the top of the building is 10,89 m.


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Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is x , Point A is 5,2 m away from the foot of the building, and the line-of-sight distance from Point A to the top of the building is 10,39 m.

Calculate the height of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The height of the building is m.
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Use the Theorem of Pythagoras to calculate the height of the building.


STEP: Use the Theorem of Pythagoras to calculate the height of the building
[−2 points ⇒ 0 / 2 points left]

The question gives us two sides of a right-angled triangle, and we need to find the third side: we can use the Theorem of Pythagoras.

c2=a2+b2(10,39)2=x2+(5,2)2Solving: x=8,99511...9 m

Therefore, the height of the building is 9 m.


Submit your answer as:

Solving two dimensional problems

A person stands at Point A, looking up at a bird sitting on the top of a building, Point B.

The height of the building is 9 m, Point A is x  away from the foot of the building, and the line-of-sight distance from Point A to the top of the building is 14,0 m.

Calculate the distance away from the foot of the building (x) as shown in the diagram below:

INSTRUCTION: Round your answer to two decimal places.
Answer: The distance away from the foot of the building is m.
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Use the Theorem of Pythagoras to calculate the distance away from the foot of the building.


STEP: Use the Theorem of Pythagoras to calculate the distance away from the foot of the building
[−2 points ⇒ 0 / 2 points left]

The question gives us two sides of a right-angled triangle, and we need to find the third side: we can use the Theorem of Pythagoras.

c2=a2+b2(14,0)2=(9)2+x2Solving: x=10,72380...10,73 m

Therefore, the distance away from the foot of the building is 10,73 m.


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